Conic Sections Questions

We provide conic sections practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on conic sections skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

Conic Sections Questions

List of conic sections Questions

Question NoQuestionsClass
134.
Tangent and normal are drawn at P(16, 16) on the parabola
y2 = 16x , which intersect the axis of the parabola at A and B,
respectively. IfC is the centre of the circle through the points
P, A and B and ZCPB = 0, then a value of tan o is:
[JEE M 2018]
(a) 2 (b) 3 (c) (d) =
11
2The graph of the equation ( 4 y^{2}+x^{2}= )
25 is
A . a circle
B. an ellipse
c. a hyperbola
D. a parabola
E . a straight line
11
3The eccentricity of the hyperbola ( x y=4 )
is
( A cdot 2 )
B. ( sqrt{2} )
( c cdot frac{3}{2} )
D. ( sqrt{3} )
11
4Find the coordinates of the foci, the
vertices, the length of major axis, the
minor axis, the eccentricity and the length of the latus rectum of the ellipse ( frac{x^{2}}{49}+frac{y^{2}}{36}=1 )
11
5The foci of the ellipse ( frac{x^{2}}{16}+frac{y^{2}}{b^{2}}=1 ) and the hyperbola ( frac{x^{2}}{144}-frac{y^{2}}{81}=frac{1}{25} ) coincide
Then, the value of ( b^{2} ) is :
A . 5
B. 7
c. 9
D.
11
6Which of the following equations
represents parametrically, parabolic
profile?
( mathbf{A} cdot x=3 cos t ; y=4 sin t )
B ( x^{2}-2=-cos t ; y=4 cos ^{2} frac{t}{2} )
( mathbf{c} cdot sqrt{x}=tan t ; sqrt{y}=sec t )
D. ( x=sqrt{1-sin t} ; y=sin frac{t}{2}+cos frac{t}{2} )
11
7Length of the latusrectum of the
hyperbola ( boldsymbol{x} boldsymbol{y}=boldsymbol{c}, ) is equal to
A ( .2 c )
B. ( sqrt{2} c )
c. ( 2 sqrt{2} c )
D. ( 4 c )
11
8Find the value of ( p ) when the parabola
( y^{2}=4 p x ) goes through the point ( (i)(3,- )
2) and (ii) (9,-12)
11
9The equation of the circle passing
through (3,6) and whose centre is (2,-1) is
A ( cdot x^{2}+y^{2}-4 x+2 y=45 )
B . ( x^{2}+y^{2}-2 y+45=0 )
c. ( x^{2}+y^{2}+4 x-2 y=45 )
D. ( x^{2}+y^{2}+2 y+45=0 )
11
10( P ) is a point on the ellipse having (3,4) and (3,-2) as the ends of minor axis. If the sum of the focal distances of ( boldsymbol{P} ) be
equal to 10 then its equation is
A ( cdot frac{(x-3)^{2}}{36}+frac{(y-1)^{2}}{12}=1 )
в. ( frac{(x-3)^{2}}{36}+frac{(y-1)^{2}}{25}=1 )
c. ( frac{(x-3)^{2}}{25}+frac{(y-1)^{2}}{9}=1 )
D. ( frac{(x-3)^{2}}{16}+frac{(y-1)^{2}}{7}=1 )
11
11The equation of the circle passing through the point (-1,2) and having two diameters along the pair of lines ( x^{2}-y^{2}-4 x+2 y+3=0 ) is
A ( cdot x^{2}+y^{2}-4 x-2 y+5=0 )
B ( cdot x^{2}+y^{2}+4 x+2 y-5=0 )
c. ( x^{2}+y^{2}-4 x-2 y-5=0 )
D. ( x^{2}+y^{2}+4 x+2 y+5=0 )
11
12Find the equation of parabola with vertex (0,0)( & ) focus at (0,2)11
13For the given parabola find the coordinates of focus, axis, the equation of the directrix and the length of the
latus rectum.
( boldsymbol{x}^{2}=-mathbf{9} boldsymbol{y} )
11
14The eccentricity of the conic ( x^{2}+ )
( 2 y^{2}-2 x+3 y+2=0 ) is
( mathbf{A} cdot mathbf{0} )
B. ( frac{1}{sqrt{2}} )
( c cdot frac{1}{2} )
D. ( sqrt{2} )
( E )
11
15( S_{1}, S_{2} ) are foci of an ellipse of major axis
of length 10 units and ( P ) is any point on the ellipse such that perimeter of
triangle ( P S_{1} S_{2} ) is ( 15 . ) Then eccentricity
of the ellipse is:
A . 0.5
B. 0.25
c. 0.28
D. 0.75
11
16Find the length of the latus rectum, the eccentricity and the coordinates of the foci of the ellipse ( x^{2}+3 y^{2}=a^{2} )11
17Let ( S ) and ( S^{prime} ) be two foci of the ellipse ( frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1 . ) If a circle described on
( S S^{prime} ) as diameter intersects the ellipse
in real and distinct points, then the eccentricity ( e ) of the ellipse satisfies
( ^{A} cdot e=frac{1}{sqrt{2}} )
в. ( _{e} inleft(frac{1}{sqrt{2}}, 1right) )
( ^{mathrm{c}} cdot_{e} inleft(0, frac{1}{sqrt{2}}right) )
D. None of these
11
18The centre of the ellipse ( 4 x^{2}+y^{2}- )
( 8 x+4 y-8=0 ) is
A ( .(0,2) )
в. (2,-1)
c. (2,1)
()
D. (1,2)
11
19In an ellipse, the distance between its focii is 6 and minor axis is ( 8 . ) Then its’
eccentricity is
A ( cdot frac{3}{5} )
B. ( frac{1}{2} )
( c cdot frac{4}{5} )
D. ( frac{1}{sqrt{5}} )
11
20A series of concentric ellipses
( epsilon_{1}, epsilon_{2}, epsilon_{3}, cdots in_{n} ) are drawn such that ( epsilon_{n} )
touches the extremities of the major
axis of ( epsilon_{n-1} ) and the focii of ( epsilon_{n} ) coincides
with the extremities of the minor axis of
( epsilon_{n-1} ) if the ( e ) of the ellipse is independent on ( n, ) then the value of ( e ? )
11
21For the ellipse ( 12 x^{2}+4 y^{2}+24 x- )
( mathbf{1 6} boldsymbol{y}+mathbf{2 5}=mathbf{0} )
A. centre is (-1,2)
B. Length of axes are ( sqrt{3} a n d 1 )
( ^{mathrm{c}} cdot_{text {eceentricity is }} sqrt{frac{2}{3}} )
D. All of these
11
22The equation ( 3 x^{2}-2 x y+y^{2}=0 )
represents:
A . a circle
B. hyperbola
c. a pair of lines
D. none of these
11
23A thin rod of length ( l ) in the shape of a semicircle is pivoted at one of its ends such that it is free to oscillate in its own
plane. The frequency ( boldsymbol{f} ) of small oscillations of the semicircular rod is :
A ( cdot frac{1}{2 pi} sqrt{frac{g pi}{2 l}} )
в. ( frac{1}{2 pi} sqrt{frac{g sqrt{pi^{2}+4}}{2 l}} )
c. ( frac{1}{2 pi} sqrt{frac{g sqrt{pi+2}}{l}} )
D. ( frac{1}{2 pi} sqrt{frac{g sqrt{pi^{2}+1}}{2 pi l}} )
11
24A circle is described with minor axis of
the ellipse as diameter. If the foci lie on the circle, then the eccentricty of the ellipse is
A ( cdot frac{1}{sqrt{3}} )
в. ( frac{1}{sqrt{2}} )
( c cdot frac{1}{2} )
D. ( frac{1}{sqrt{5}} )
11
25les 2x – 3y=5 and 3x – 4y=7 are diameters of a circle
of area 154 sq. units. Then the equation of this circle is
(a) x2 + y2 + 2x – 2y=62
(1989 – 2 Marks)
(b) x2 + y2 + 2x – 2y=47
(C) x2 + y2 – 2x +2y=47
(d) x2 + y2– 2x + 2y =62
11
26Find the coordinates of the focus axis of
the parabola the equation of directrix and the length of the latus rectum for ( y ) ( 2=10 x )
11
27The area of an ellipse is ( 8 pi ) sq. units. Its distance between the foci is ( 4 sqrt{3} ), then
( mathbf{e}= )
( mathbf{A} cdot sin 30^{circ} )
B. ( sin 45^{circ} )
c. ( sin 60^{circ} )
( mathbf{D} cdot sin 75^{circ} )
11
28An ellipse having foci at (3,1) and
(1,1) passes through the point (1,3)
Its eccentricity is:
A ( cdot sqrt{2}-1 )
B. ( sqrt{3}-1 )
c. ( frac{1}{2}(sqrt{2}-1) )
D. ( frac{1}{2}(sqrt{3}-1) )
11
2934.
.X _Y_=1 is inscribed in a rectangle R
The ellipse E1 : +
9xyz
whose sides are parallel to the coordinate axes. Another
ellipse E, passing through the point (0,4) circumscribes the
rectangle R. The eccentricity of the ellipse E2 is (2012)
11
30The centre of circle inscribed in square formed by
scribed in square formed by the lines
x2 – 8x + 12 =0 and y2 – 14y+45 = 0, is
(20035)
(a) (4,7) (b) (7,4) (c) (9,4) (d) (4,9)
11
31Find the coordinates of the foci, the
vertices the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse ( frac{x^{2}}{4}+frac{y^{2}}{25}=1 )
11
32Which of the following is/are not false?
This question has multiple correct options
A. The mid point of the line segment joining the foci is called the centre of the ellipse.
B. The line segment through the foci of the ellipse is called the major axis.
C. The end points of the major axis are called the vertices of the ellipse.
D. Ellipse is symmetric with respect to Y-axis only.
11
33For the given parabola find the coordinates of focus, axis, the equation of the directrix and the length of the
latus rectum.
( boldsymbol{x}^{2}=mathbf{1 6} boldsymbol{y} )
11
3427. The normal to the curve, x2 + 2xy – 3y2 = 0, at(1,1)
(JEE M 2015]
(a) meets the curve again in the third quadrant.
(b) meets the curve again in the fourth quadrant.
(c) does not meet the curve again.
(d) meets the curve again in the second quadrant.
11
35I.
:
Find the equation of the circle whose radius is 5 and which
touches the circle x2 + y2 – 2x – 4y-20=0 at the point (5,5).
(1978)
11
36If ( t ) is a parameter,then ( boldsymbol{x}=boldsymbol{a}left(boldsymbol{t}+frac{mathbf{1}}{boldsymbol{t}}right) )
and ( y=bleft(t-frac{1}{t}right) ) represent
A. An ellipse
B. A circle
c. A pair of straight lines
D. A hyperbola
11
3733. Tangents are drawn to the hyperbola 4×2 – y2 = 36 at the
points P and Q. If these tangents intersect at the point
T(0, 3) then the area (in sq. units) of APTQ is :
[JEEM 2018]
(a) 5413 (b) 6013 (©) 3615 (d) 4515
11
38Latus rectum of a parabola is a ( ldots ldots . . ) line
segment with respect to the axis of the parabola through the focus whose endpoints lie on the parabola.
A. perpendicular
B. parallel
c. tilted
D. None of these
11
39Find equation of latus rectum of the
parabola ( (x+1)^{2}=32 y )
A ( . y=32 )
B . ( x-8=0 )
( mathbf{c} cdot y-8=0 )
D. ( x=32 )
11
40-Topic-wal D
14. Let P and Q be distinct points
Let P and Q be distinct points on the parabola y = 2x such
nat a circle with PO as diameter passes through the vertex
O of the parabola. If P lies in the first quadrant and the area
of the triangle AOPQ is 3
11
41Find the eccentricity of an ellipse, if its
latus rectum be equal to one half its
minor axis.
11
42Find the equation of the ellipse in the standard form whose distance between
foci is 2 and the length of latus rectum is ( frac{15}{2} )
11
4325.
be the circle with centre at (1, 1) and radius = 1. If Tis
the circle centred at (0, y), passing through origin and
touching the circle C externally, then the radius of T is equal
to
[JEE M 2014]
(a)
11
44Find the locus of a point which moves in
such a way that the sum of its distances from(4,3) and (4,1) is 5.
11
45An ellipse passing through the point ( (2 sqrt{13}, 4) ) has its foci at (-4,1) and ( (4,1), ) then its eccentricity is
A ( cdot frac{2}{3} )
B. ( frac{1}{3} )
( c cdot frac{1}{4} )
D.
11
46T
U
VUIUUS UIP
21.
The circle x2 + y2 = 4x+8y+5 intersects the line 3x – 4y=m
at two distinct points if
[2010]
(a) – 35<m<15
(b) 15<m<65
(c) 35<m<85
(d) -85<m<-35
2
2
2
O
11
47Find the co-ordinates of the point from
which tangents drawn to the circle ( x^{2}+y^{2}-6 x-8 y+3=0 ) such that the
mid point of its chord of contact is (1,1)
11
48Find the equation of centre (1,1) and radius ( sqrt{2} )11
49Find the equation of the following curve in cartesian form ( boldsymbol{x}=-mathbf{1}+mathbf{2} sin boldsymbol{theta}, boldsymbol{y}= )
( 1+2 cos theta . ) find the centre and radius
of circle.
11
5015.
A focus of an ellipse is at the origin. The directrix is the line
x = 4 and the eccentricity is
Then the length of the
semi-major axis is
[2008]
11
51The latus rectum of the hyperbola ( 16 x^{2}-9 y^{2}=144 ) is
A ( cdot frac{13}{6} )
в. ( frac{32}{3} )
( c cdot frac{8}{3} )
D. ( frac{4}{3} )
11
5232. Let P(6,3) be a point on the hyperbola
5 =1. If the
al
normal at the point P intersects the x-axis at (9,0), then the
eccentricity of the hyperbola is
(2011)
(c)
2
(d)
3
11
5317. Normals are drawn
Normals are drawn from the point P with slopes m 1, m,m,
to the parabola y2 = 4x. If locus of P with m, m2 = a is a part
of the parabola itself then find a. (2003 – 4 Marks)
4
ot
:
11
54A variable point ( boldsymbol{P} ) on the ellipse of eccentricity is joined to the foci ( S ) and
( s^{prime} . ) The eccentricity of the locus of the in
cetre of the triangle ( P S S^{1} ) is
A ( cdot sqrt{frac{2 e}{1+e}} )
B. ( sqrt{frac{e}{1+e}} )
c. ( sqrt{frac{1-e}{1+e}} )
D. ( frac{e}{2(1+e)} )
11
55For the parabola ( boldsymbol{y}^{2}+mathbf{8} boldsymbol{x}-mathbf{1 2} boldsymbol{y}+mathbf{2 0}= )
( mathbf{0} )
This question has multiple correct options
A. vertex is (2,6)
B. Focus is (0.6)
c. Latusrectum 4
D. Axis ( y=6 )
11
56For an ellipse, ( A ) and ( B ) are the ends of major axis and minor axis respectively. Area of ( Delta O A B ) is 16 sq.units and ( e= ) ( frac{sqrt{3}}{2}, ) then equation of the ellipse is:
( stackrel{text { A }}{-} frac{x^{2}}{32}+frac{y^{2}}{8}=1 )
в. ( frac{x^{2}}{16}+frac{y^{2}}{64}=1 )
c. ( frac{x^{2}}{64}+frac{y^{2}}{8}=1 )
D. ( frac{x^{2}}{64}+frac{y^{2}}{16}=1 )
11
57Find the equation of the hyperbola whose foci are ( (0, pm sqrt{10}) ) and passing through the point (2,3)11
58( S ) and ( T ) are the foci of an ellipse and ( B )
is an end of the minor axis. If ( S T B ) is an
equilateral triangle, then what is the value of ( e ) ?
A ( cdot frac{1}{4} )
B. ( frac{1}{3} )
( c cdot frac{1}{2} )
D. ( frac{2}{3} )
11
59The lines ( 2 x-3 y=5 ) and ( 3 x-4 y=7 )
intersect at the center of the circle
whose area is 154 sq. units, then equation of circle is
A ( cdot x^{2}+y^{2}-2 x+2 y=47 )
B. ( x^{2}+y^{2}+2 x-2 y=31 )
c. ( x^{2}+y^{2}-2 x-2 y=47 )
D. ( x^{2}+y^{2}-2 x-2 y=31 )
11
608.
Let L, be a strainght line passing through the origin and L,
be the straight line x +y = 1. If the intercepts made by the
circle x2 + y2 – x+3y = 0 on L, and L, are equal, then
which of the following equations can represent L,?
(1999 – 3 Marks
(a) x+y=0
(b) x-y=0
(c) x+7y=0
(d) x-7y = 0
11
61Which of the following is/are correct?
This question has multiple correct options
A. Parabola is symmetric with respect to the axis of the parabola.
B. Length of latus rectum of a parabola, ( y^{2}=4 a x ) is ( 4 a )
C. A line through the focus and perpendicular to the directrix is called the axis of the parabola.
D. The point of intersection of a parabola with the axis is called the vertex of the parabola.
11
6214. If the pair of lines ax2 +2 (a + b)xy + by
of lines ar? +2 (a + b)xy + by2 = 0 lie along
imeters of a circle and divide the circle into four sectors
such that the area of one of the sectors is thrice the a
another sector then
[2005]
(a) 3a2 – 10ab + 362 = 0 (b) 3a2 – 2ab + 362 = 0
(©) 3a+10ab + 362 = 0 (d) 3a? + 2ab + 362 = 0
11
632
le passes through the point (a, b) and cuts the circle
2 + y2 = 4 orthogonally, then the locus of its centre is
(a) 2ax – 2by – (a² +6²+4)= 0
[2004]
(b) 2ax + 2by – (a? +b+ 4) = 0
(C) 2ax – 2by +(a? + b2 + 4) = 0
(d) 2ax + 2by + (a? +62 + 4) = 0
11
64In an ellipse the length of minor axis is equal to the distance between the foci,
the length of latus rectum is 10 and ( e=frac{1}{sqrt{2}} . ) Then the length of semi major
axis is:
A . 16
B . 18
c. 10
D. 22
11
65A point ( (alpha, beta) ) lies on a circle ( x^{2}+y^{2}= )
1, then locus of the point ( (3 alpha+2 beta) ) is a
/ an.
A. Straight line
B. Ellipse
c. Parabola
D. None of these
11
66Find the equations of the circle passing through (4,3) and
touching the lines x + y = 2 and x-y=2. (1982-3 Marks)
11
67Find the direction cosines of the unit
vector perpendicular to the plane ( vec{r} cdot(6 hat{i}-3 hat{j}-2 k)+1=0 ) passing
through the origin.
11
68Find the equation of the hyperbola whose Transverse and Conjugate axes
are the ( x ) and ( y ) axes respectively, given
that the length of conjugate axis is 5 and distance between the foci is 13
11
6926. Let O be the vertex and Q be any point on the parabola,
x = 8y. If the point P divides the line segment OQ internally
in the ratio 1:3, then locus of P is: [JEE M 2015]
(a) y2 = 2x (b) x2 = 2y (c) x2 =y (d) y2 = X
11
70The total number of real tangents that
can be drawn to the ellipse ( 3 x^{2}+ ) ( 5 y^{2}=32 ) and ( 25 x^{2}+9 y^{2}=450 )
passing through (3,5) is
( A cdot 0 )
B . 2
( c .3 )
D.
11
7119.
ition of the family of circles with fixed
The differential equation of the family of circie
radius 5 units and centre on the line y=2 is
(a) (x – 2)y2 = 25+(y-2)2
(b) (y-2)y’2 = 25-(y-2)2
(c) (-2)?y2 = 25-(y-2)2
(d) (x – 2)2 y2 = 25 -(y-2)2
11
725.
A circle S passes through the point (0, 1) and is orthogonal
to the circles (x – 1)2 + y2 = 16 and x2 + y2 = 1. Then
(JEE Adv. 2014)
(a) radius of S is 8 (b) radius of Sis 7.
(c) centre of Sis (-7, 1) (d) centre of S is (-8,1)
11
73The centre, vertex, focus of a conic are
( (0,0),(0,5),(0,6) . ) Its length of latus rectum is
A ( cdot frac{11}{5} )
в. ( frac{7}{5} )
c. ( frac{14}{5} )
D. ( frac{22}{5} )
11
74The centre of a circle passing through the points (0,0), (1,
and touching the circle x2 + y2 =9 is (1992 – 2 Marks)
11
75Prove that the sum of the distances
from the focus of the points in which a conic is intersected by any circle, whose centre is at a fixed point on the transverse axis, is constant.
11
76Through a fixed point (h, k) secants are drawn to the circle
x2 + y2 = r2. Show that the locus of the mid-points of the
secants intercepted by the circle is x2 + y2 = hx + ky.
(1983 – 5 Marks)
11
77The eccentricity of the conic ( 9 x^{2}+ )
( 5 y^{2}-54 x-40 y+116=0 ) is:
A ( -frac{1}{3} )
B. ( frac{2}{3} )
( c cdot frac{4}{9} )
D. ( frac{2}{sqrt{5}} )
11
78The locus of the centre of a circle, which touches externany
the circle x2 + y2 – 6x – 6y + 14 = 0 and also touch
is, is given by the equation: (1993 – 1 Marks)
(a) x2 – 6x – 10y + 14 =0 (b) x2 – 10x -6y + 14 = 0
(C) y2 – 6x – 10y + 14 =0 (d) y2 – 10x – 6y + 14 = 0
2.
et
11
79The point at which the hyperbola
intersects the transverse axis are called
the ( ldots ). of the hyperbola.
11
80Find the eccentricity of that ellipse,
whose latus rectum is half of the minor
axis.
11
81Find the coordinates of the point at which the circles
x + y – 4x – 2y= 4 and x2 + y2 – 12x – 8y=-36 touch each
other. Also find equations common tangents touching the
circles in the distinct points.
(1993 – 5 Marks)
11
82The ends of the latus rectum of the
parabola ( boldsymbol{x}^{2}+mathbf{1 0 x}-mathbf{1 6 y}+mathbf{2 5}=mathbf{0} ) are
A ( cdot(3,4),(-13,4) )
B. (5,-8),(-5,8)
c. (3,-4),(13,4)
D. (-3,4),(13,-4)
11
838.
The circle x2 + y2 – 4x-4y+4=0 is inscribed in a triangle
which has two of its sides along the co-ordinate axes. The
locus of the circumcentre of the triangle is
x + y – xy + k(x2 + y2)1/2 = 0. Find k. (1987- 4 Marks)
11
8417.
The area of the quadrilateral formed by the tangents at the
end points of latus rectum to the ellipse –
5
(a) 27/4 sq. units
(c) 27/2 sq. units
(b) 9 sq. units
(d) 27 sq. units
(2003)
11
852.
IfP=(x, y), F, =(3,0), F, =(-3,0) and 16×2 + 25y2 = 400, then
PF, +PF, equals
(1998 – 2 Marks)
(a) 8 (6) 6 (c) 10 (d) 12
11
86The equation ( 16 x^{2}-3 y^{2}-32 x+ )
( 12 y-44=0 ) represents a hyperbola.
A. The length of whose transverse axis is ( 4 sqrt{3} )
B. The length of whose conjugate axis is 4
c. whose centre is (-1,2)
D. whose eccentricity is ( sqrt{frac{19}{3}} )
11
8712. Find the co-ordinates of all the points P on the ellipse
r2 y2
5 =1, for which the area of the triangle PON is
maximum, where O denotes the origin and N, the foot of the
perpendicular from O to the tangent at P. (1999 – 10 Marks)
11
88Let ( A B C D ) be a square of side length 1
and ( Gamma ) a circle passing through ( B ) and ( C ) and touching ( A D . ) The radius of ( Gamma ) is
A
в.
c. ( frac{1}{sqrt{2}} )
D.
11
89The eccentricity of the ellipse ( 25 x^{2}+ )
( 16^{2}=400 ) is
A ( cdot frac{3}{5} )
B. ( frac{1}{3} )
( c cdot frac{2}{5} )
D.
11
90Suppose that the normals drawn at three different points on
the parabola y2 = 4x pass through the point (h, k). Show
that h>2.
(1981 – 4 Marks)
11
912.
The normal at the point (bt,2,2bt,) on a parabola meets
the parabola again in the point (bt22, 2btą), then
(b) t2 =4-2 [2003]
11
(d) 12=1-
11
92The locus of a point P(a, b) moving under the condition
that the line y = ax +ß is a tangent to the hyperbola
x2 12
[2005]
a² 62
(a) an ellipse
c) a parabola
(b) a circle
(d) a hyperbola
11.
11
93lines 2x – 3y = 5 and 3x – 4y = 7 are diameters of a
having area as 154 sq.units. Then the equation of the
[2003]
(a) x2 + y2 – 2x +2y = 62 (b) x2 + y2 + 2x – 2y = 62.
(C) x2 + y2 + 2x – 2y = 47. (d) x2 + y2 – 2x + 2y = 47
circle is
11
11
94The equation of the latusrecta of the ellipse ( 9 x^{2}+4^{2}-18 x-8 y-23=0 )
are
A ( . y=pm sqrt{5} )
В. ( x=pm sqrt{5} )
c. ( y=1 pm sqrt{5} )
D. ( x=1 pm sqrt{5} )
11
95If the eccentricities of the hyperbola ( frac{x^{2}}{a^{2}}-frac{y^{2}}{b^{2}}=1 ) and ( frac{y^{2}}{b^{2}}-frac{x^{2}}{a^{2}}=1 ) be ( e ) and
( e_{1}, ) then ( frac{1}{e^{2}}+frac{1}{e_{1}^{2}}= )
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot 3 )
D. None of these
11
96Find the equation of the ellipse whose
vertices are ( (pm mathbf{3}, mathbf{0}) ) and foci are
(±2,0)
11
9731. A hyperbola passes through the point P(V2,V3) and
has foci at (+2,0). Then the tangent to this hyperbola at
P also passes through the point :
[JEEM 2017]
(a) (-12,-v3 (b) (312,213)
(0) (2/2, 3/3)
(2) (13, 12)
11
98The equation of the image of the circle ( x^{2}+y^{2}+16 x-24 y+183=0 ) by the
line mirror ( 4 x+7 y+13=0 ) is:
A ( cdot x^{2}+y^{2}+32 x-4 y+235=0 )
B. ( x^{2}+y^{2}+32 x+4 y-235=0 )
C. ( x^{2}+y^{2}+32 x-4 y-235=0 )
D. ( x^{2}+y^{2}+32 x+4 y+235=0 )
11
99If the line ( y=x sqrt{3}-3 ) cuts the
parabola ( boldsymbol{y}^{2}=boldsymbol{x}+boldsymbol{2} ) at ( boldsymbol{P} ) and ( boldsymbol{Q} ) and if
( A ) be the points ( (sqrt{3}, 0), ) then ( A P . A Q ) is
A ( cdot frac{2}{3}(sqrt{3}+2) )
B. ( frac{4}{3}(sqrt{3}+2) )
c. ( frac{4}{3}(2-sqrt{3}) )
D. ( frac{4}{6-3 sqrt{3}} )
11
1006.
Lines 5x+12y-10=0 and 5x – 12y-40 = 0 touch a circle C
of diameter 6. If the centre of C, lies in the first quadrant,
find the equation of the circle C, which is concentric with
C, and cuts intercepts of length 8 on these lines.
11
101Find the coordinates of the foci, the
vertices the eccentricity and the length
of latus rectum of the hyperbola ( 9 y^{2}- ) ( 4 x^{2}=36 )
11
10214.
The triangle PQR is inscribed in the circle x2 + y2 = 25. If o
and R have co-ordinates (3,4) and (4,3) respectively, then
ZQPR is equal to
(2000)
11
103TOPIL
21.
The angle between the tangents drav
s drawn from the point (1.4)
to the parabola y2 = 4x is
(a) TUG (6) N4 (c) T3 (d) T2
(2004)
11
104The focus of a conic is the origin and its
corresponding directrix is ( 7 x-y- )
( mathbf{1 0}=mathbf{0 .} ) If length of its latus rectum is 2
then its eccentricity is:
A ( cdot sqrt{2} )
B. 1
c. ( frac{1}{sqrt{2}} )
D. ( frac{1}{2} )
11
105


10.
The angle between a pair of tangents drawn from a point P
to the parabola y2 =4ax is 45°. Show that the locus of the
point P is a hyperbola.
(1998 – 8 Marks)
11
106A straight line through the vertex P of a triangle POR
intersects the side OR at the point S and the circumcircle of
the triangle PQR at the point T. If S is not the centre of the
circumcircle, then
(2008)
2
11
(a) ps sīTOS XSR (6) PS ST TOS XSR
1 1,4
1 1 4
© PS + STOR (d) PS+ STOR
11
107Given below are Matching type questions, with two columns(each
having some items) each.Each item of column ( I ) has to be matched with the
items of column ( I I ), by enclosing the
correct match.
Note:An item of column ( I ) can be
matched with more than one items of
columnII.All the items of column ( I I )
have to be matched. The equation of conics represented by the general equation of second degree ( boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{2 h x} boldsymbol{y}+boldsymbol{b} boldsymbol{y}^{2}+boldsymbol{2} boldsymbol{g} boldsymbol{x}+boldsymbol{2} boldsymbol{f} boldsymbol{y}+boldsymbol{c}= )
0
and the discriminant of above equation is represented by ( triangle, ) where ( triangle=a b c+ ) ( 2 f g h-a f^{2}-b g^{2}-c h^{2} ) or ( left(begin{array}{lll}a & h & g \ h & b & f \ g & f & cend{array}right) )
Now, match the entries from the
following two columns
11
108The equation ( frac{x^{2}}{8-t}+frac{y^{2}}{t-4}=1 ) will
represent an ellipse if
A. ( t in(1,5) )
B. ( t in(2,8) )
c. ( t in(4,8)-{6} )
D. ( t in(4,10)-{6} )
11
10913. Ifa circle passes through the point (a, b) and cuts the circle
x2 + y2 = p< orthogonally, then the equation of the locus
of its centre is
[2005]
(a) x2 + y2 – 3ax – 4by +(a? +62 – p2)=0
(b) 2ax +2by-(a? – b? +p?)=0
(C) x² + y2 – 2ax – 3by +(a? – b2 – p2)=0
(d) 2ax +2by-(a? +62 +p)=0
11
110The equation of the ellipse whose equation of directrix is ( 3 x+4 y-5=0 )
coordinates of the focus are (1,2) and the eccentricity is ( frac{1}{2} ) is ( 91 x^{2}+84 y^{2}- )
( 24 x y-170 x-360 y+475=0 )
A. True
B. False
11
111An ellipse with centre at (0,0) cuts ( x ) axis at (3,0) and ( (-3,0) . ) If its ( e=frac{1}{2} ) then the length of the semiminor axis is:
A ( .2 sqrt{3} )
B. ( sqrt{5} )
c. ( 3 sqrt{2} )
D. ( frac{3 sqrt{3}}{2} )
11
112Find the latus rectum, the eccentricity, and the coordinates of the foci, of the
ellipses
(1) ( x^{2}+3 y^{2}=a^{2}, ) (2) ( 5 x^{2}+4 y^{2}=1 )
and
(3) ( 9 x^{2}+5 y^{2}-30 y=0 )
11
113The tangent at any point ( P(a cos theta, b sin theta) ) on the ellipse ( frac{x^{2}}{a^{2}}+ )
( frac{boldsymbol{y}^{2}}{boldsymbol{b}^{2}}=1 ) meets the auxiliary circle at two
point which subtend a right angle at the centre, then eccentricity is
A ( cdot frac{1}{sqrt{1+sin ^{2} theta}} )
в. ( frac{1}{sqrt{2-cos ^{2} theta}} )
c. ( frac{1}{sqrt{1+tan 2 theta}} )
D. noneofthese
11
11468. The radii of two circles are 5cm
and 3cm, the distance between
their centres is 24 cm. Then
the length of the transverse
common tangent is
(1) 16 cm (2) 15.2 cm
(3) 162
cm (4) 15 cm
11
11520. If a chord, which is not a tangent, of the parabola y2 = 16x
has the equation 2x +y=p, and midpoint (h, k), then which
of the following is(are) possible value(s) of p, h and k?
(JEE Adv. 2017)
(a) p=-2, h=2, k=-4
(b) p=-1, h=1, k=-3
(c) p=2, h=3, k=-4
(d) p=5, h=4, k=-3
11
116If ( L_{1} L_{2} ) is the latusrectum of ( y^{2}=12 x )
( P ) is any point on the directrix then the
( operatorname{area~of} Delta P L_{1} L_{2}= )
( A cdot 32 )
B . 18
( c .36 )
D. 16
11
117The length of the latus rectum of the parabola
( mathbf{1 6 9}left{(boldsymbol{x}-mathbf{1})^{2}+(boldsymbol{y}-mathbf{3})^{2}right}=(mathbf{5} boldsymbol{x}- )
( 12 y+17)^{2} ) is
A ( cdot frac{14}{11} )
в. ( frac{12}{13} )
c. ( frac{28}{13} )
D. none of these
11
118If the the hyperbola ( frac{x^{2}}{4}-frac{y^{2}}{b^{2}}=1 )
passses though (4,3)
( mathbf{A} cdot b^{2}=3 )
B. ( b^{2}=9 )
( mathbf{c} cdot b^{2}=4 )
D. ( b^{2}=100 )
11
119Find the equation to the circle :
Whose radius is ( sqrt{a^{2}-b^{2}} ) and whose
center is ( (-a,-b) )
11
120The length of latus rectum of ( frac{x^{2}}{9}+ ) ( frac{boldsymbol{y}^{2}}{2}=1 ) is
( A cdot frac{7}{4} )
B. ( frac{3}{4} )
( c cdot frac{4}{3} )
D. None.
11
121a)
1
(0)
14. If a > 26 > 0 then the positive value of m for which
y = mx – 671+ m2 is a common tangent to
x2 + y2 = 52 and (x – a)2 + y2 = 62 is
(20025)
26
Va? – 462
(b)
=
2b
©
26
a-26
a-2b
11
122( S ) is one focus of an ellipse and ( P ) is any point on the ellipse. If the maximum and minimum values of ( S P ) are ( m ) and
( boldsymbol{n} ) respectively, then the length of semi
major axis is
A . AM of ( m, n )
в. G.M. of ( m, n )
с. нм от ( m, n )
D. AGP of ( m, n )
11
123The equation of the latus rectum of the hyperbola ( frac{(x-4)^{2}}{16}-frac{(y-3)^{2}}{20}=1 )
are?
( mathbf{A} cdot x=1 pm 5 )
B . ( x=4 pm 6 )
c. ( y=2 pm 6 )
D. ( y=3 pm 5 )
11
124The locus of the vertices of the family of parabolas
[2006]
11
125Find the coordinates of the foci, the
vertices, the length of major axis, the
minor axis, the eccentricity and the length of the latus rectum of the ellipse ( frac{x^{2}}{36}+frac{y^{2}}{16}=1 )
11
126Find the equation of the circle whose
diameters are along the lines ( 2 x- )
( 3 y+12=0 ) and ( x+4 y-5=0 ) and
whose area is 154 sq. units.
11
127If (5,12) and (24,7) are the foci of an
ellipse passing through the origin, then the eccentricity of the conic is
A ( cdot frac{sqrt{386}}{12} )
B. ( frac{sqrt{386}}{13} )
c. ( frac{sqrt{386}}{25} )
D. ( frac{sqrt{386}}{38} )
11
128If latus rectum of an ellipse ( frac{x^{2}}{16}+ ) ( frac{boldsymbol{y}^{2}}{boldsymbol{b}^{2}}=1{0<boldsymbol{b}<4}, ) subtends angle ( mathbf{2} boldsymbol{theta} )
at farthest vertex such that ( operatorname{cosec} theta= )
( sqrt{5} ) then which of the following options
are correct:
This question has multiple correct options
A ( cdot e=frac{1}{2} )
B. no such ellipse exists
c. ( b=2 sqrt{3} )
D. area of ( Delta ) formed by ( L R ) and nearest vertex is 6 sq units
11
12915.
The locus of the mid-point of the line segment joining the
focus to a moving point on the parabola y2 = 4ax is another
parabola with directrix
(2002)
(a) x=-a (b) x=-a/2 (c) X=0 (d) x=a/2
11
130The AFC Curve passes through the Origin statement is –
A. True
B. False
c. Partially True
D. Nothing can be said
11
131If the line x – 1 = 0 is the directrix of the parabola
ye – kx + 8 = 0, then one of the values of k is (
(a) 1/8 (b) 8
(c) 4
(d) 1/4
1: the circle
11
132For the points on the circle ( x^{2}+y^{2}- )
( 2 x-2 y+1=0, ) the sum of maximum
and minimum values of ( 4 x+3 y ) is
A ( cdot frac{26}{3} )
B . 10
c. 12
D. 14
11
133Through a fixed point (h, k) secants are drawn to the circle
x2 + y2 = r2. Show that the locus of the mid-points of the
secants intercepted by the circle is x2 + y2 = hx + ky.
(1983 – 5 Marks)
11
1345.
(1994 – 4 MUTAS)
Show that the locus of a point that divides a chord of slope
2 of the parabola y2 = 4x internally in the ratio 1: 2 is a
parabola. Find the vertex of this parabola. (1995 – 5 Marks)
of the
11
135The length of the latus rectum of the
parabola ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}^{2}+boldsymbol{b} boldsymbol{y}+boldsymbol{c} ) is
A ( cdot frac{a}{4} )
B. ( frac{a}{3} )
( c cdot frac{1}{a} )
D. ( frac{1}{4 a} )
11
136Find the foci of the curve ( 16 x^{2}- )
( mathbf{2 4 x y}+mathbf{9 y}^{2}+mathbf{2 8 x + 1 4 y}+mathbf{2 1}=mathbf{0} )
11
137The intercept on the line ( y=x ) by the
( operatorname{circle} x^{2}+y^{2}-2 x=0 ) is ( A B )
Equation of the circle with ( A B ) as a
diameter is
A ( cdot x^{2}+y^{2}+x+y=0 )
B . ( x^{2}+y^{2}-x-y=0 )
c. ( x^{2}+y^{2}+x-y=0 )
D. None of these
11
138.
64
4
10.
In an ellipse, the distance between its foci is 6 and minor
axis is 8. Then its eccentricity is
[2006]
(d)
11
13922. If the line 2x + V6 y=2 touches the hyperbola x2 – 2y =4,
then the point of contact is
(2004)
(a) (-2, 0)
(b) (-5,216)
(d) (4,-16)
11
140The locus of a planet orbiting around the sun is:
A. A circle
B. A straight line
c. A semicircle
D. An ellipse
11
1415. The axis of a parabola is along the line y=x and the distances
of its vertex and focus from origin are 12 and 22
respectively. If vertex and focus both lie in the first quadrant.
then the equation of the parabola is (2006 – 3M, -1)
(a) (x + y)2 = (x -y-2) (b) (x – y)2 = (x + y-2)
© (x – y)2 = 4 (x + y -2) (d) (x – y)2 = 8 (x + y-2)
11
142[2004]
If the lines 2x + 3y +1 = 0 and 3x – y -4 = 0 lie along
diameter of a circle of circumference 10t, then the equation
of the circle is
(a) x² + y2 + 2x – 2y – 23 = 0
(b) x2 + y2 – 2x – 2y – 23 = 0
© x2 + y2 + 2x+2y – 23 = 0
(d) x² + y2 –2x+2y – 23 = 0
11
143The eccentricity of the conic
represented by the equation ( x^{2}+ )
( 2 y^{2}-2 x+3 y+2=0 ) is
( mathbf{A} cdot mathbf{0} )
B. ( frac{1}{2} )
c. ( frac{1}{sqrt{2}} )
D. ( sqrt{2} )
11
14439.
Equation of a common tangent to the circ
x2 + y2 – 6x = 0 and the parabola, y2 = 4X, 18:
[JEEM 2019-9 Jan (M)
(a) 2 V3y=12x +1
(C) 2 V3 y=-x-12
(b) 13y=x+3
(d) 13 y=3x +1
11
145The graph represented by equations ( x=sin ^{2} t, y=2 cos t ) is
A. hyperbola
B. sine graph
c. parabola
D. straight line
11
146Find the coordinates of the foci, the
vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse ( 4 x^{2}+9 y^{2}=36 )
11
14738.
Let 0 < 0
. If the eccentricity of the hyperbola
r2 v2
cos?o sin’e
20 = 1 is greater than 2, then the length of its
latus rectum lies in the interval: JEE M 2019-9 Jan (M)
(a) (3,0)
(b) (3/2,2]
(C) (2,31
(d) (1,3/2]
11
148The tangents at two points, ( P ) and ( Q ), of a conic meet in ( mathrm{T} ), and ( mathrm{S} ) is the focus
prove that if the conic be a parabola,
( operatorname{then} S T^{2}=S P . S Q )
11
149Length of the latus rectum of the parabola ( 25left[(x-2)^{2}+(y-3)^{2}right]= )
( (3 x-4 y+7)^{2} ) is :
( A cdot 4 )
B. 2
( c cdot 1 / 5 )
D. 2/5
11
150If the length of the major axis of the ellipse ( left(frac{x^{2}}{a^{2}}right)+left(frac{y^{2}}{b^{2}}right)=1 ) is three
times the length of minor axis, its eccentricity is:
A ( cdot frac{1}{3} )
в. ( frac{1}{sqrt{3}} )
( c cdot sqrt{frac{2}{3}} )
D. ( frac{2 sqrt{2}}{3} )
11
151Find the coordinates of the foci, the
vertices, the length of major axis, the minor axis, the eccentricity and the
length of the latus rectum of the ellipse ( 36 x^{2}+4 y^{2}=144 )
11
1526.
Let P(x, y) and Q(x, y,), y, <0,y, <0, be the end points of
the latus rectum of the ellipse x2 + 4y2 = 4. The equations of
parabolas with latus rectum PQ are
(2008)
(a) x2 +213 y = 3+v3 (b) -213 y = 3+ 13
(C) x2 + 273 y = 3 -13 (d) x2 – 2/3 y = 3-13
11
153For ( boldsymbol{y}^{2}-mathbf{2 0 0} boldsymbol{x}=mathbf{0}, ) focal distance of a
point ( (2,20), ) is
A . 48
B. 50
c. 52
D. 20
11
1548.
If x=9 is the chord of contact of the hyperbola x2 – y2 = 9,
then the equation of the corresponding pair of tangents is
(1999 – 2 Marks)
(a) 9×2 – 8y2 + 18x –9=0 (b) 9×2 – 892 – 18x +9=0
(c) 9×2 – 8y2 – 18x –9=0 (d) 9×2 – 872 +18x+9=0
11
155State whether following statements are
true or false
Statement-1: The only circle having
radius ( sqrt{10} ) and a diameter along line
( 2 x+y=5 ) is ( x^{2}+y^{2}-6 x+2 y=0 )
Statement-2: The line ( 2 x+y=5 ) is a
normal to the circle ( x^{2}+y^{2}-6 x+ )
( 2 y=0 )
A. Statement- 1 is false, statement- 2 is true.
B. Statement-1 is true,statement-2 is true and statement 2 is NOT the correct explanation for statement-1.
c. Statement-1 is true, statement-2 is false.
D. Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-
11
156Find the area of the partitions cut off by the hyperbola ( x^{2}-3 y^{2}=1 ) from the
ellipse ( x^{2}+4 y^{2}=8 )
11
157If the equation ( 136left(x^{2}+y^{2}right)=(5 x+ )
( 3 y+7)^{2} ) represents a conic, then its
length of latus rectum is
A ( cdot frac{7}{2 sqrt{34}} )
B. ( frac{7}{sqrt{34}} )
c. ( frac{14}{sqrt{34}} )
D. ( frac{9}{sqrt{34}} )
11
158( S ) and ( T ) are the foci of an ellipse and ( B )
is an end of the minor axis, if ( triangle boldsymbol{S T B} ) is
equilateral, then ( e ) is equals to:
A ( cdot frac{1}{4} )
B. ( frac{1}{3} )
( c cdot frac{1}{2} )
D. None of these
11
159( boldsymbol{E}_{1}=frac{boldsymbol{x}^{2}}{boldsymbol{a}^{2}}+frac{boldsymbol{y}^{2}}{boldsymbol{b}^{2}}-mathbf{1}=mathbf{0},(boldsymbol{a}>boldsymbol{b}) ) and
( boldsymbol{E}_{2}=frac{boldsymbol{x}^{2}}{boldsymbol{k}^{2}}+frac{boldsymbol{y}^{2}}{boldsymbol{b}^{2}}-mathbf{1}=mathbf{0},(boldsymbol{k}<boldsymbol{b}) boldsymbol{E}_{2} ) is
inscribed in ( E_{1} . ) If ( E_{1} ) and ( E_{2} ) have same
eccentricities then length of minor axis
of ( boldsymbol{E}_{2}=boldsymbol{p}left(mathrm{LLR} text { of } boldsymbol{E}_{1}right) ) then ( boldsymbol{p}=? )
A ( cdot frac{1}{2} )
B. 1
( c cdot frac{2}{3} )
D.
11
160The asymptotes of a hyperbola having centre at the point (1,2) are parallel to the lines ( 3 x+4 y=0 ) and ( 4 x+5 y=0 )
If the hyperbola passes through the point ( (3,5), ) Find the equation of the hyperbola.
11
161The equation of the circle which
touches ( x ) -axis and whose center is (1,2)
is
( mathbf{A} cdot x^{2}+y^{2}-2 x+4 y+1=0 )
B . ( x^{2}+y^{2}-2 x-4 y+1=0 )
( mathbf{c} cdot x^{2}+y^{2}+2 x+4 y+1=0 )
D. ( x^{2}+y^{2}-4 x+2 y+4=0 )
11
162If the centre ( O ) of circle is the
intersection of ( x- ) axis and line ( y= ) ( frac{4}{3} x+4, ) and the point (3,8) lies on
circle, then the equation of circle will be
A ( cdot x^{2}+y^{2}=25 )
B – ( (x+3)^{2}+y^{2}=25 )
c. ( (x+3)^{2}+y^{2}=100 )
D. ( (x+3)^{2}+(y-8)^{2}=100 )
11
163Find the equation of the ellipse referred to its centre whose minor axis is equal to the distance between the foci and
whose latus rectum is 10 .
11
164The eccentricity of the conic is ( frac{l}{r}= ) ( 2+3 cos theta+4 sin theta )
( mathbf{A} cdot mathbf{3} )
B. 4
( c .5 )
D.
11
16518. The point diametrically opposite to the point
P(1,0) on the circle x2 + y2 + 2x + 4y -3=0 is
[2008]
(a) (3,-4) (b) (-3,4) (c) (-3,-4) (d) (3,4)
11
166A hyperbola, having the transverse axis of length 2 sin , je
confocal with the ellipse 3×2 + 4y2 = 12. Then its equation is
(2007 – 3 marks)
(a) x?cosec20-y-sec20 = 1 (b) x?sec20-y-cosec20 =1
(C) x-sin20- y2cos20=1 (d) x?cos20-yasin20=1
11
167The length of latus rectum of the
hyperbola ( boldsymbol{x} boldsymbol{y}-mathbf{3} boldsymbol{x}-mathbf{3} boldsymbol{y}+mathbf{7}=mathbf{0} ) is
A .4
B. 3
( c cdot 2 )
D.
11
16816. The equation of the common tangent to the curves yz = 8x
and xy=-1 is
(a) 3y=9x + 2
(b) y=2x + 1
(c) 2y=x+8
(d) y=x+2
(2002)
11
16920. Let 2×2 + y2 – 3xy = 0 be the equation of a pair of tangents
drawn from the origin O to a circle of radius 3 with centre in
the first quadrant. If A is one of the points of contact, find
the length of OA.
(2001 – 5 Marks
11
1703.
Consider the two curves Cy: y2 = 4x, C2: x2 + y2 – 6x +1=0.
Then,
(2008)
1 C, and C, touch each other only at one point.
in C, and C, touch each other exactly at two points
C, and C, intersect (but do not touch) at exactly two
points
(d). C, and C, neither intersect nor touch each other
11
171A man running round a race course notes that the sum of the distance of
two flag posts from him is always ( 10 m ) and the distance between the flag posts
is 8 m. The area of the path he encloses
in square metres is:
A . ( 15 pi )
в. ( 20 pi )
c. ( 27 pi )
D. ( 30 pi )
11
172(1992- O MURS)
Consider a family of circles passing through two fixed points
A (3,7) and B(6,5). Show that the chords in which the circle
x + y2 – 4x – 6y – 3 = 0 cuts the members of the family are
concurrent at a point. Find the coordinate of this point.
11
173Find the equation of the circle:
Centered at (3,-2) with radius 4
( mathbf{A} cdot x^{2}+y^{2}+6 x-4 y=3 )
B ( cdot x^{2}+y^{2}-6 x+4 y=3 )
C ( cdot x^{2}+y^{2}-3 x+2 y=-3 )
D. ( x^{2}+y^{2}+3 x-2 y=-3 )
11
17421. Let C, and C, be two circles with C, lying inside C. A circle
C lying inside C, touches C, internally and C, externally.
Identify the locus of the centre of C. (2001 – 5 Marks)
11
175If the latus rectum of an ellipse ( boldsymbol{x}^{2} tan ^{2} varphi+boldsymbol{y}^{2} sec ^{2} varphi=1 ) is ( 1 / 2, ) then ( varphi )
is
( mathbf{A} cdot pi / 2 )
в. ( pi / 6 )
c. ( pi / 3 )
D. ( 5 pi / 12 )
11
17621. An ellipse is drawn by taking a diameter of the circle (x – 1)
+ y2 = 1 as its semi-minor axis and a diameter of the circle
x2+(y-2)2=4 is semi-major axis. If the centre of the ellipse
is at the origin and its axes are the coordinate axes, then the
equation of the ellipse is :
[2012]
(a) 4×2 + y2 = 4
(b) x2 + 4y2=8
(c) 4×2 + y2 = 8
(d) x2 + 4y2 = 16
11
177Cantres of the three circles
( x^{2}+y^{2}-4 x-6 y-14=0 )
( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{2} boldsymbol{x}+boldsymbol{4} boldsymbol{y}-boldsymbol{5}=mathbf{0} )
and ( x^{2}+y^{2}-10 x-16 y+7=0 )
A. Are the vertices of a right triangle
B. The vertices of an isosceles triangle which is not regular
c. vertices of a regular triangle
D. Are collinear
11
178Find the coordinates of the point of
intersection of the axis and the directrix
of the parabola whose focus is (3,3) and directrix is ( 3 x-4 y=2 . ) Find also
the length of the latus-rectum.
11
1792.
Two circles x2 + y2 =6 and x2 + y2 – 6x +8=0 are given. Then
the equation of the circle through their points of intersection
and the point (1,1) is
(1980)
(a) x2 + y2 – 6x +4=0 (b) x2 + y2 – 3x + 1 = 0
(c) x2 + y2 – 4y +2=0 (d) none of these
11
180UNTUV 1 14
0
(u
10
11.
The circles x2 + y2 – 10x + 16 = 0 and x2 + y2 = pe in
each other in two distinct points if
(a) r8 (c) 2<r<8 (d) 25r58
(1994)
11
181The set of points ( (x, y) ) whose distance from the line ( y=2 x+2 ) is the same as
the distance from (2,0) is a parabola. This parabola is congruent to the
parabola in standard form ( y=K x^{2} ) for
some ( K ) which is equal to
A ( frac{sqrt{5}}{12} )
в. ( frac{sqrt{5}}{4} )
c. ( frac{4}{sqrt{5}} )
D. ( frac{12}{sqrt{5}} )
11
182The differential equation ( (3 x+4 y+ )
1) ( boldsymbol{d} boldsymbol{x}+(boldsymbol{4} boldsymbol{x}+boldsymbol{5} boldsymbol{y}+mathbf{1}) boldsymbol{d} boldsymbol{y}=boldsymbol{0} ) represents
a family of
A. Circles
B. Parabolas
c. Ellipses
D. Hyperbolas
11
183Draw the circles whose equation are
( 3 x^{2}+3 y^{2}=4 x )
11
18446. Chords AB and CD of a circle
intersect externally at P. If AB
= 6 cm, CD = 3 cm and PD = 5
cm, then the length of PB is
(1) 5 cm (2) 6.25 cm
(3) 6 cm (4) 7.35 cm
11
185The graph of ( x^{2}-4 x+y^{2}+6 y=0 ) in
the xy-plane is a circle. What is the
radius of the circle?
11
186( frac{x^{2}}{r^{2}-r-6}+frac{y^{2}}{r^{2}-6 r+5}=1 ) will
represents the ellipse, if r lies in the interval:
This question has multiple correct options
B. ( (3, infty) )
( c cdot(5, infty) )
D. ( (1, infty) )
11
187If the latus rectum of an ellipse ( boldsymbol{x}^{2} tan ^{2} varphi+boldsymbol{y}^{2} sec ^{2} varphi=1 ) is ( 1 / 2 ) then ( varphi )
is
( mathbf{A} cdot pi / 2 )
в. ( pi / 6 )
c. ( pi / 3 )
D. ( 5 pi / 12 )
11
188The centre of a circle is (2,-3) and the circumference is ( 10 pi . ) Then, the equation of the circle is
A. ( x^{2}+y^{2}+4 x+6 y+12=0 )
B . ( x^{2}+y^{2}-4 x+6 y+12=0 )
c. ( x^{2}+y^{2}-4 x+6 y-12=0 )
D. ( x^{2}+y^{2}-4 x-6 y-12=0 )
11
189General second degree equation in ( x ) and ( y ) is ( a x^{2}+2 h x y+b y^{2}+2 g x+ )
( 2 f y+c=0, ) Where ( a, h, b, g, f ) and ( c ) are
constats.

Prove that condition for it to be a circle
is: ( a=b ) and ( h=0 )

11
190he centre of the circle passing through (0,0) and (1,0) and
touching the circle x2+y2=9 is
[2002]
(a) (31) (b) (2.-13) (21) (a) (2)
mation of
11
191The axis of the parabola ( x^{2}-4 x-y+ )
( mathbf{1}=mathbf{0} ) is
A ( cdot y=-3 )
B. x=-3
c. ( x=2 )
D. none of these
11
192The eccentricity of the ellipse ( 9 x^{2}+ )
( 25 y^{2}-18 x-100 y-116=0, ) is
A ( .25 / 16 )
в. ( 4 / 5 )
c. ( 16 / 25 )
D. ( 5 / 4 )
11
193The equation of the latus rectum of the parabola ( boldsymbol{x}^{2}+mathbf{4} boldsymbol{x}+mathbf{2} boldsymbol{y}=mathbf{0} ) is –
( A cdot 3 y=2 )
B. ( 2 y+3=0 )
c. ( 2 y=3 )
D. ( 3 y+2=0 )
11
194Find the equation of the ellipse whose
vertices are ( (pm mathbf{5}, mathbf{0}) ) and foci are
( (pm mathbf{3}, mathbf{0}) )
11
195If the eccentricity of an ellipse is ( frac{5}{8} ) and the distance between its foci is ( 10, ) then
its latus rectum is
A ( cdot frac{39}{4} )
B. 12
c. 15
D. ( frac{37}{2} )
11
1966.
Let P be the point (1,0) and Q a point on the locus y = 8x.
The locus of mid point of PQ is
[2005]
(a) y2 – 4x+2=0 (b) y2 +4x+2=0
(C) x² +4y+2=0 (d) x2 – 4y+2=0
11
197If ( e_{1} ) and ( e_{2} ) are the eccentricities of ( a )
hyperbola ( 3 x^{2}-3 y^{2}=25 ) and its
conjugate respectively, then
A ( cdot e_{1}^{2}+e_{2}^{2}=2 )
B . ( e_{1}^{2}+e_{2}^{2}=4 )
( mathbf{c} cdot e_{1}+e_{2}=4 )
D. ( e_{1}+e_{2}=sqrt{2} )
11
198Equation of the parabola whose vertex is (0,0) and focus is the point of intersection of the lines ( boldsymbol{x}+boldsymbol{y}= )
( 2,2 x-y=4 ) is
A ( cdot y^{2}=2 x )
в. ( y^{2}=4 x )
c. ( y^{2}=8 x )
D . ( x^{2}=8 y )
11
199If the lines ( 2 x+3 y+1=0 ) and ( 3 x- )
( mathbf{y}-mathbf{4}=mathbf{0} ) lie along diameters of a circle
of circumference ( 10 pi, ) then the
equation of the circle is:
A ( cdot x^{2}+y^{2}-2 x+2 y-23=0 )
B. ( x^{2}+y^{2}-2 x-2 y-23=0 )
C. ( x^{2}+y^{2}+2 x+2 y-23=0 )
D. ( x^{2}+y^{2}+2 x-2 y-23=0 )
11
2001
Let a circle be given by 2x(x -a) + y(2y-b)=0,(a + 0,
5 0 ).
Find the condition on a and b if two chords, each bisected
by the x-axis, can be drawn to the circle from
11
201Find the equation of the parabola that satisfies the following conditions: Focus (0,-3)( ; ) directrix ( y=3 )11
202The difference between the length ( 2 a ) of the transverse axis of a hyperbola of
eccentricity ( e ) and the length of its latus
rectum is :
A ( cdot 2 aleft|3-e^{2}right| )
В ( cdot 2 aleft|2-e^{2}right| )
c. ( 2 aleft(e^{2}-1right) )
D. ( aleft(2 e^{2}-1right) )
11
203-*
The locus of the mid-point of the chord of contact of
tangents drawn from points lying on the straight line 4x – 5y
= 20 to the circle x2 + y2 = 9 is
(2012)
(a) 20(x2 + y2) – 36x +45 y=0
(b) 20 (x2 + y2) + 36x – 45 y=0
(C) 36(x2 + y2) – 20x +45 y
11
204State whether the following statements
are true or false.
The equation ( x^{2}+y^{2}+2 x-10 y+ )
( mathbf{3 0}=mathbf{0} ) represents the equation of a
circle.
A . True
B. False
11
205If a circle passes through the point (a, b) and cuts the circle
x + y2 = k orthogonally, then the equation of the locus of
its centre is
(1988 – 2 Marks)
(a) 2ax + 2by – (a? +62 +62) = 0
(b) 2ax + 2by – (a? – b2 + k_) = 0
(C) x2 + y2 – 3ax – 4by+(a? +62 – K2) = 0
(d) x2 + y2 – 2ax – 3by +(22 -62 – k) = 0.
11
206What is the eccentricity of the conic ( 4 x^{2}+9 y^{2}=144 )
A ( frac{sqrt{5}}{3} )
в. ( frac{sqrt{5}}{6} )
c. ( frac{3}{sqrt{5}} )
D. ( frac{2}{3} )
11
207Find the equation of a circle if:
(i)center ( (a, b) ) and radius ( sqrt{a^{2}+b^{2}} )
(ii)center ( (a sec alpha, b tan alpha) ) and radius
( sqrt{a^{2} sec ^{2} alpha+b^{2} sec ^{2} alpha} )
11
208Find the equation of the ellipse whose
foci are ( (mathbf{0}, pm mathbf{6}) ) and length of the minor
axis is 16
11
209( frac{x^{2}}{8-a}+frac{y^{2}}{a-2}=1 ) represents an
ellipse.Then find range of ‘a’
11
210The equation of parabola whose latus rectum is 2 units, axis is ( x+y-2=0 )
and tangent at the vertex is ( x-y+ )
( 4=0 ) is given by
A ( cdot(x+y-2)^{2}=4 sqrt{2}(x-y+4)^{2} )
B . ( (x-y-4)^{2}=4 sqrt{2}(x+y-2) )
C. ( (x+y-2)^{2}=2 sqrt{2}(x-y+4) )
D. ( (x-y-4)^{2}=2 sqrt{2}(x-y+2)^{2} )
11
211The equation of the hyperbola whose foci are ( (mathbf{6}, mathbf{5}),(-mathbf{4}, mathbf{5}) ) and eccentricity ( mathbf{5} / mathbf{4} ) is?
A ( cdot frac{(x-1)^{2}}{16}-frac{(y-5)^{2}}{9}=1 )
В. ( frac{x^{2}}{16}-frac{y^{2}}{9}=1 )
c. ( frac{(x-1)^{2}}{16}-frac{(y-5)^{2}}{9}=-1 )
D. ( frac{(x-1)^{2}}{4}-frac{(y-5)^{2}}{9}=1 )
11
212The the focal distance of an end of the
minor axis of any ellipse (reffered to its axes of ( x ) and ( y ) respectively) is ( k ) and
the distance between the foci is ( 2 h )
then its equation is:
A ( cdot frac{x^{2}}{k^{2}}+frac{y^{2}}{k^{2}+h^{2}}=1 )
B. ( frac{x^{2}}{k^{2}}+frac{y^{2}}{h^{2}-k^{2}}=1 )
( ^{mathrm{C}} cdot frac{x^{2}}{k^{2}}+frac{y^{2}}{k^{2}-h^{2}}=1 )
D. ( frac{x^{2}}{k^{2}}+frac{y^{2}}{h^{2}}=1 )
11
213Find the locus of coordinates of ( A ) and ( B )
which are two points in place so that
( boldsymbol{P A}-boldsymbol{P B}= ) constant.
11
214Find the eccentricity and length of latus rectum of the ellipse ( 9 x^{2}+16 y^{2}- )
( 36 x+32 y-92=0 )
11
215In parabola ( y^{2}=18 x ) find the point
where the ordinate is equal to three times the abscissa.
11
216The graph of the equation ( x^{2}+frac{y^{2}}{4}=1 )
is
A. an ellipse
B. a circle
c. a hyperbola
D. a parabola
E. two straight lines
11
217A circle is given by x2 + (y-1)2 = 1, another circle C touches
it externally and also the x-axis, then the locus of its centre is
(2005S)
(a) {(x, y): x2 = 4y} {(x,y): y=0}
(b) {(x, y): x2 + (y – 1)2 = 4}U {(x, y): y 50}
(c) {(x, y): x2=y} U{(0,y):y>0}
(d) {(x, y): x2 = 4y} {(0,y):y s0}
11
218Find the equation of parabola with focus (5,0) and vertex (5,3)
A ( cdot y=12(x-5)^{2}-3 )
B . ( y-3=-(x-5)^{2} )
c. ( y=frac{(x-5)^{2}}{12}-3 )
D. ( y=-frac{(x-5)^{2}}{12}+3 )
11
219Find the equation of the hyperbola satisfying the give conditions: Vertices
( (pm mathbf{7}, mathbf{0}) )
( e=frac{4}{3} )
11
220Coordinates of the focus of the parabola
( x^{2}-4 x-8 y-4=0 ) are
A ( .(0.2) )
B. (2,1)
c. (1,2)
D. (-2,-1)
11
221The length of latus rectum of the hyperbola ( 4 x^{2}-9 y^{2}-16 x-54 y- )
( mathbf{1 0 1}=mathbf{0} ) is
( A cdot frac{8}{5} )
B. ( frac{8}{7} )
( c cdot frac{8}{9} )
D. ( frac{8}{3} )
11
222Find the equation of the ellipse whose
vertices are ( (pm mathbf{7}, mathbf{0}) ) and foci are
( (pm mathbf{4}, mathbf{0}) )
11
223The foci of the ellipse ( frac{x^{2}}{16}+frac{y^{2}}{b^{2}}=1 ) and the hyperbola ( frac{x^{2}}{144}-frac{y^{2}}{81}=frac{1}{25} ) coincide
then the value of ( b^{2} ) is:
A . 5
B. 7
c. 9
D. 4
11
224The length of the latus rectum of the ( operatorname{conic} frac{5}{r}=3-2 cos theta ) is
A ( cdot frac{3}{5} )
B. ( frac{5}{3} )
( c cdot frac{6}{5} )
D. ( frac{10}{3} )
11
225Find the centre and radius of the circle
( x^{2}+y^{2}=36 )
11
226The equation of the circle which touches the lines ( x=0, y=0 ) and
( 4 x+3 y=12 ) is
A ( cdot x^{2}+y^{2}-2 x-2 y-1=0 )
В . ( x^{2}+y^{2}-2 x-2 y+3=0 )
c. ( x^{2}+y^{2}-2 x-2 y+2=0 )
D. ( x^{2}+y^{2}-2 x-2 y+1=0 )
E . ( x^{2}+y^{2}-2 x-2 y-3=0 )
11
227Match the Column
The answers to these questions have to be appropriately doubled
11
228Find the equation of the ellipse whose
foci are ( (mathbf{0}, pm mathbf{6}) ) and length of the minor
axis is 22
11
229Find the lengths of, and the equations to, the focal radii drawn to the point ( (4 sqrt{3}, 5) ) of the ellipse ( 25 x^{2}+16 y^{2}= )
( mathbf{1 6 0 0} )
11
230Find the equation of the hyperbola satisfying the give conditions: Vertices
(0,±5) foci (0,±8)
11
231The equation of the latus rectum of the ellipse ( 9 x^{2}+4 y^{2}-18 x-8 y-23=0 )
are
A ( . y=pm sqrt{5} )
B ( cdot y=-sqrt{5} )
c. ( y=1 pm sqrt{5} )
D. ( y=-1 pm sqrt{5} )
11
232The equation ( sqrt{(x-2)^{2}+y^{2}}+ )
( sqrt{(x+2)^{2}+y^{2}}=5 ) represents
A . a circle
B. ellipse
c. line segment
D. an empty set
11
233If the two circles (x – 1)2 + (y – 3)2 = r- and
x + y2 – 8x +2y +8=0 intersect in two distinct po
sect in two distinct points, then
(1989- 2 Marks)
(a)
2<r<8 (b)
r2
1
11
234An ellipse with foci (2,2),(3,-5)
passes through ( (6,-1), ) then its semilatus rectum is:
( A cdot frac{7}{2} )
в. ( frac{5}{2} )
( c cdot frac{9}{2} )
D. ( frac{11}{2} )
11
235Find vertex and focus for the equation
( boldsymbol{y}^{2}-mathbf{8} boldsymbol{y}-boldsymbol{x}+mathbf{1} boldsymbol{9}=mathbf{0} )
11
2364
17. Consider a family of circles which
tamily of circles which are passing through the
point (1, 1) and are tangent to x-axis. ”
coordinate of the centre of the circles, then the set ol
of k is given by the interval
[2007]
tangent to x-axis. If (h, k) are the
of the circles, then the set of values
(a)
— Sk
(b) ks
VIN
(C)
0 <k
(d) kz
11
237The equation of the hyperbola with vertices (0,±15) and foci (0,±20) is
A ( cdot frac{x^{2}}{175}-frac{y^{2}}{225}=1 )
В. ( frac{x^{2}}{625}-frac{y^{2}}{125}=1 )
c. ( frac{y^{2}}{225}-frac{x^{2}}{125}=1 )
D. ( frac{y^{2}}{225}-frac{x^{2}}{175}=1 )
11
238Consider a rigid square ( A B C D ) as in
the figure with ( A ) and ( B ) on the ( x ) and ( y )
axis respectively. When ( A ) and ( B ) slide
along their respective axes, the locus of
( C ) forms a part of
A. A circle
B. A parabola
C. A hyperbola
D. An ellipse which is not a circle
11
239O and 2x – 3y-5=0 are two
Stt Square units, the equation
[2006]
15. If the lines 3r-47-7=0 and 2x – 3y –
diameters of a circle of area 491 square un
of the circle is
(a) x2 + y2 + 2x – 2y – 47 = 0
(b) x2 + y2 + 2x – 2y – 62 = 0
(c) x2 + y2 – 2x + 2y -62 = 0
(d) x² + y2 –2x+2y – 47 = 0
6. La
11
240Find the length of the latus rectum, the eccentricity and the coordinates of the foci of the ellipse
( mathbf{5} boldsymbol{x}^{2}+mathbf{4} boldsymbol{y}^{2}=mathbf{1} )
11
241The angle between a pair of tangents drawn from a point P
to the circle x2 + y2 + 4x – 6y +9 sin? a +13 cos2 a=0 is 2a.
The equation of the locus of the point Pis
(1996 – 1 Mark)
(a) x2 + y2 + 4x – 6y + 4 =0 (b) x2 + y2 + 4x – 6y-9=0
(C) x2 + y2 + 4x – 6y-4=0 (d) x2 + y2 + 4x – 6y +9=0
11
242Which ordered number pair represents the center of the circle ( x^{2}+y^{2}-6 x+ )
( 4 y-12=0 ? )
( A cdot(9,4) )
B. (3,2)
( c cdot(3,-2) )
( D cdot(6,4) )
11
243Find the eccentricity of an ellipse in which distance between the foci is 10
and that of focus and
corresponding directrix is 15
A ( cdot frac{1}{4} )
B. ( frac{1}{2} )
c. 1
D. ( frac{3}{4} )
11
244Let P be the point on the parabola, y = 8x W
Which
on the parabola, y2 = 8x which is at a
ce from the centre Cofthe circle, x2 +(y+6 =1.
n of the circle, passing through Cand having
[JEEM 2016]
& the
minimum distance from the centre
Then the equation of the circle, passin
its centre at P is:
(a) x² + y2 -*+2y-24 =0
(b) x2 + y2 – 4x +9y+18=0
(c) x2 + y2-4x+8y+12=0
(d) x2 + y2 -x+4y-12=0
11
245Find the focus of the parabola ( boldsymbol{y}= )
( -2(x+4)^{2}-1 )
( mathbf{A} cdot(-4,-1) )
в. ( left(-4, frac{9}{8}right) )
c. (-4,1)
D. ( left(-4,-frac{9}{8}right) )
11
246Find the equation of the ellipse whose
vertices are ( (pm mathbf{9}, mathbf{0}) ) and foci are
( (pm mathbf{5}, mathbf{0}) )
11
2477.
Let a given line L, intersects the x and y axes at P and Q.
respectively. Let another line L, perpendicular to L, cut
the x and y axes at R and S, respectively. Show that the
locus of the point of intersection of the lines PS and QR is a
circle passing through the origin (1987- 3 Marks)
11
24816. A parabola has the origin as its focus and the line x = 2 as
the directrix. Then the vertex of the parabola is at [2008]
(a) (0,2) (b) (1,0) (c) (0,1) (d) (2,0)
11
249The sum of the distances of any point on the ellipse ( 3 x^{2}+4 y^{2}=24 ) from its foci
is :
( A cdot 8 sqrt{2} )
B. 8
c. ( 16 sqrt{2} )
D. ( 4 sqrt{2} )
11
250If the latusrectum of an ellipse is equal to half of minor axis, find its
eccentricity.
11
25115.
If the circles x2 + y2 + 2x + 2ky + 6 = 0
ne circles x2 + y2 + 2x + 2ky+6=0. x2 + y2 + 2ky + k=0
intersect orthogonally, then k is
(20005)
(2) 2017 (6) 2012 (1) 2017 / (2) – 208 2 / 2
11
252( 2 x^{2}+y^{2}-8 x-2 y+1=0 )
Find the square of the Latus Rectum for
the given ellipse.
11
2537.
Let a given line L, intersects the x and y axes at P and Q,
respectively. Let another line L2, perpendicular to L,, cut
the x and y axes at R and S, respectively. Show that the
locus of the point of intersection of the lines PS and QR is a
circle passing through the origin. (1987 – 3 Marks)
11
25436.
If the tangent at (1, 7) to the curve x2 = y-6 touches the
circle x² + y2 +16x +12y+c=0 then the value of cis :
JEEM 2018
(a) 185 (b) 85 (3) 95 (4) 195
11
255The equation ( frac{x^{2}}{10-a}+frac{y^{2}}{4-a}=1 )
represents an ellipse if
( mathbf{A} cdot a4 )
c. ( 4<a<10 )
D. None of these
11
256entres of a set of circles, each of radius 3, lie on the
12+12=25. The locus of any point in the set is
(b) x2+y2 <25
[2002]
(c) x²+ y = 25 (d) 3 < x² + y² <9
chironi
6
(a) 4<x²+77 < 64
11
257Eccentricity of the ellipse ( 4 x^{2}+y^{2}-8 y-8=0 ) is
A ( frac{sqrt{3}}{2} )
в. ( frac{sqrt{3}}{4} )
( c cdot frac{sqrt{3}}{sqrt{2}} )
D. ( frac{sqrt{3}}{8} )
E ( cdot frac{sqrt{3}}{16} )
11
258A straight line drawn through the common focus ( mathrm{S}^{prime} ) of a number of conics
meets them in the points ( P_{1}, P_{2}, ldots ; ) on
it is taken a point ( Q ) such that the reciprocal of ( mathrm{SQ} ) is equal to the sum of
the reciprocals of ( S P_{1}, S P_{2}, ldots ) Prove that the locus of ( Q ) is a conic section
whose focus is ( 0, ) and show that the reciprocal of its latus rectum is equal to
the sum of the reciprocals of the latera
recta of the given conics.
11
259Find the inclinations of the axes so that
the following equations may represent circles, and in each case find the radius
and centre;
( boldsymbol{x}^{2}-boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}-2 boldsymbol{g} boldsymbol{x}-boldsymbol{2} boldsymbol{f} boldsymbol{y}=mathbf{0} )
11
260A parabola has ( x ) – axis as its axis, ( y ) axis as its directrix and ( 4 a ) as its latus
rectum. If the focus lies to the left side
of the directrix then the equation of the parabola is
( mathbf{A} cdot y^{2}=4 a(x+a) )
B ( cdot y^{2}=4 a(x-a) )
C ( cdot y^{2}=-4 a(x+a) )
D ( cdot y^{2}=4 a(x-2 a) )
11
261For the given parabola find the coordinates of focus, axis, the equation of the directrix and the length of the
latus rectum.
( x^{2}=6 y )
11
262The order of the differential equation of the family of parabolas whose length of latus rectum is fixed and axis is the ( x )
axis
A .2
B. 1
( c .3 )
D. 4
11
263The number of common tangents to the circles x2 + y2 = 4
and x2 + y2 – 6x — 8y= 24 is
(1998 – 2 Marks)
(a) o (6) 1 (c) 3 (d) 4
11
264In an ellipse the distance between the foci is one third of the distance between
the directrices, then its ( e ) is
A ( cdot frac{1}{2} )
B. ( frac{1}{sqrt{3}} )
( c cdot frac{2 sqrt{2}}{3} )
D. ( frac{1}{3} )
11
265If
(2,4) and (10,10) are the ends of a latus – rectum of an ellipse with eccentricity ( frac{1}{2}, ) then the length of semi major axis is
A ( cdot frac{20}{3} )
в. ( frac{15}{3} )
c. ( frac{40}{3} )
D. None of these
11
2661.
Find the equation of the circle whose radius is 5 and which
touches the circle x2 + y2 – 2x – 4y-20=0 at the point (5,5).
(1978)
11
267The graph between ( log left(theta-theta_{0}right) ) and time ( (t) ) is a straight line in the experiment based on Newton’s law cooling. What is the shape of graph between ( theta ) and ( t ? )
A. A straight line
B. A parabola
c. A hyperbola
D. A circle
11
26852. The distance between the centres
of two circles of radii 6 cm and 3
cm is 15 cm. The length of the
transverse common tangent to
the circles is :
(1) 12 cm (2) 6.76 cm
(3) 7/6 cm (4) 18 cm
11
269IS of a parabola lies along x-axis. If its vertex and focus
are at distance 2 and 4 respectively from
positive x-axis then which of the following points does not
lie on it?
JEE M 2019-9 Jan (ML
(a) (5,256)
(b) (8,6)
(C) (6, 412)
(d) (4, -4)
11
270The asymptotes of a hyperbola ( 4 x^{2}- )
( 9 y^{2}=36 ) are
A. ( 2 x pm 3 y=1 )
в. ( 2 x pm 3 y=0 )
c. ( 3 x pm 2 y=1 )
D. None
11
271The equation of hyperbola whose coordinates of the foci are (±8,0) and
the lenght of latus rectum is 24 units, is
A ( cdot 3 x^{2}-y^{2}=48 )
в. ( 4 x^{2}-y^{2}=48 )
c. ( x^{2}-3 y^{2}=48 )
D. ( x^{2}-4 y^{2}=48 )
11
272The latus rectum of a parabola whose focal chord is ( P S Q ) such that ( S P=3 )
and ( S Q=2, ) is given by
A ( cdot frac{24}{5} )
в. ( frac{12}{5} )
( c cdot frac{6}{5} )
D. ( frac{48}{5} )
11
273The equation of circle with its centre at
the origin is ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{r}^{2} )
A. True
B. False
c. Neither
D. Either
11
274Find the equation of the ellipse whose
vertices are ( (pm mathbf{5}, mathbf{0}) ) and foci are
(±1,0)
11
27541.
If one end of a focal chord of the parabola, y =
(1.4), then the length of this focal chord is:
[JEE M 2019-9 April (M)).
(a) 25
(6) 22
(c) 24
(d) 20
11
276( P ) and ( Q ) are the foci of the ellipse ( frac{x^{2}}{a^{2}}+ ) ( frac{y^{2}}{b^{2}}=1 ) and ( B ) is an end of the minor
axis. If ( triangle P B Q ) is an equilateral triangle, the eccentricity of the ellipse is:
A ( cdot frac{1}{sqrt{2}} )
в. ( frac{1}{3} )
( c cdot frac{1}{2} )
D. ( frac{sqrt{3}}{2} )
11
277If the angle between the lines joining the end points of minor axis of an ellipse with its foci is ( frac{pi}{2}, ) then the eccentricity of the ellipse is
A ( cdot frac{1}{2} )
в. ( frac{1}{sqrt{2}} )
c. ( frac{sqrt{3}}{2} )
D. ( frac{1}{2 sqrt{2}} )
11
278The point to which the axes are to be translated to eliminate ( x ) and ( y ) terms in
the equation ( 3 x^{2}-4 x y-2 y^{2}-3 x- )
( 2 y-1=0 ) is
( ^{mathrm{A}} cdotleft(frac{5}{2}, 3right) )
в. ( left(-4, frac{3}{2}right) )
c. (-2,3)
D. (2,3)
11
279An ellipse has its centre at (1,-1) and semi-major axis ( =8 ) and it passes
through the point ( (1,3) . ) The equation of the ellipse is
A ( cdot frac{(x+1)^{2}}{64}+frac{(y+1)^{2}}{16}=1 )
в. ( frac{(x-1)^{2}}{64}+frac{(y+1)^{2}}{16}=1 )
c. ( frac{(x-1)^{2}}{16}+frac{(y+1)^{2}}{64}=1 )
D. ( frac{(x+1)^{2}}{64}+frac{(y-1)^{2}}{16}=1 )
11
280A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse
x +2y=6 at Pand Q. Prove that the tangents at P and Qof
the ellipse x2 + 2y2 = 6 are at right angles. (1997 – 5 Marks)
11
28122. The equation of the circle passing through the foci of the
ellipse
– = 1, and having centre at (0, 3) is
16
(a) x2 + y2 – 6y – 7=0
(C) x2 + y2 – 6y – 5 = 0
(JEE M 2013
(b) x2 + y2 – 6y +7=0
(d) x2 + y2 – 6y +5 = 0
11
282If ( (a, b) ) lies on circle with centre as origin, then its radius will be
A ( . a-b )
B. ( a+b )
c. ( sqrt{a^{2}+b^{2}} )
D. ( a^{2}+b^{2} )
11
283Circles are described on the major axis and the line joining the foci of the ellipse ( 3 x^{2}+2 y^{2}=6 ) as diameters.
Then the radii of the circles are in the
ratio:
A ( cdot sqrt{2}: 1 )
B. ( sqrt{3}: 1 )
( c .3: 2 )
D. 5: 4
11
28414.
I
The equation of the circle passing through (1, 1) and
points of intersection of x2 + y2 + 13x – 3y =
2×2 + 2y2 + 4x – 7y-25=0 is
(1983 – 1 Mark)
(a) 4×2 + 4y2 – 30x – 10y-25=0
(b) 4×2 + 4y2 + 30x – 13y-25 = 0
(c) 4×2 + 4y2 – 17x – 10y +25=0
(d) none of these
11
285Find the equation of the parabola with
vertex (0,0) and focus at (-3,0)
11
28611.
Consider the family of circles x2 + y2 = r2, 2<r<5. Ifin the
first quadrant, the common taingent to a circle of this family
and the ellipse 4×2 + 25y2 = 100 meets the co-ordinate axes at
A and B, then find the equation of the locus of the mid-point
of AB.
(1999 – 10 Marks)
11
28713. The equation of a tangent to the parabola
y2 = 8x is y = x + 2. The point on this line from which the
other tangent to the parabola is perpendicular to the given
tangent is
[2007]
(a) (2,4) (b) (-2,0) (c) (-1,1) (d) (0,2)
11
288Find the equation of the circle drawn on the intercept made by the line ( 2 x+ ) ( 3 y=6 ) between the coordinate axes as
diameter.
11
289Centre of the hyperbola ( x^{2}+4 y^{2}+ )
( 6 x y+8 x-2 y+7=0 ) is
A ( .(1,1) )
в. (0,2)
D. None of these
11
290A point ( P ) on the ellipse ( frac{x^{2}}{25}+frac{y^{2}}{9}=1 ) has the eccentric angle ( frac{pi}{8} . ) The sum of the distance of ( P ) from the two foci is
A . 5
B. 6
c. 10
D. 3
11
291Find the equation of the circle passing through the points (5,5),(3,7) and has its center on the line ( boldsymbol{x}-mathbf{4} boldsymbol{y}+mathbf{1 1}=mathbf{0} )11
292For an ellipse with axes are coordinate axes, ( A ) and ( L ) are the ends of major axis and latusrectum respectively. Area of ( triangle O A L=8 s q . ) units and, ( e=frac{1}{sqrt{2}}, ) then
equation of the ellipse is:
( ^{text {A }} cdot frac{x^{2}}{16}+frac{y^{2}}{8}=1 )
B. ( frac{x^{2}}{32}+frac{y^{2}}{16}=1 )
c. ( frac{x^{2}}{64}+frac{y^{2}}{32}=1 )
D. ( frac{x^{2}}{8}+frac{y^{2}}{4}=1 )
11
29318.
Tangent is drawn to parabola y2 – 2y – 4x + 5 = 0 at a point
P which cuts the directrix at the point Q. A point R is such
that it divides OP externally in the ratio 1/2 : 1. Find the
lomne afnaint D
(2011 Mantra)
11
294For the function ( f(x)=a(x-h)^{2}+k )
which of the following statements is
wrong?
A. The value of ( a ) is negative
B. ( f(x) ) is symmetrical across the line ( y=3 )
c. The function ( g(x)=frac{2 x}{3} ) intersects ( f(x) ) at its vertex
D. The value of ( h ) is positive
11
295The equation of ellipse with the length
of major and minor axis as ( 12,16 mathrm{cm} ) respectively is
11
296Which is not represented by quadratic equation?
A . Circle
B. Straight line
c. Parabola
D. Hyperbola
11
297The foci of an ellipse are ( boldsymbol{S}(-1,-1), boldsymbol{S}^{prime}(0,-2) ) and its ( mathbf{e}=frac{1}{2} )
then the equation of the directrix corresponding to the focus ( S ) is :
A. ( x-y+3=0 )
в. ( x-y+7=0 )
c. ( x-y+5=0 )
D. ( x-y+4=0 )
11
298Find the eccentricity of the conic represented by ( boldsymbol{x}^{2}-boldsymbol{y}^{2}-boldsymbol{4} boldsymbol{x}+boldsymbol{4} boldsymbol{y}+ )
( mathbf{1 6}=mathbf{0} )
A ( cdot sqrt{2} )
B. ( sqrt{3} )
( c cdot-sqrt{2} )
D. ( -sqrt{3} )
11
299The equation of a parabola which passes through the intersection of a straight line ( x+y=0 ) and the circle ( x^{2}+y^{2}+4 y=0 ) is.
A ( cdot y^{2}=4 x )
В. ( y^{2}=x )
c. ( y^{2}=2 x )
D. None of these
11
300The centre of the circle given by ( mathbf{r} cdot(mathbf{i}+ )
( 2 mathbf{j}+2 mathbf{k})=15 ) and ( |mathbf{r}-(mathbf{j}+2 mathbf{k})|=mathbf{4} )
A ( .(0,1,2) )
в. (1,3,4)
c. (-1,3,4)
D. None of these
11
30116.
) TV-2X + 2y-47 = U
Let C be the circle with centre (O,
equation of the locus of the mid points of the cho
ne circle with centre (0,0) and radius 3 units. The
he mid points of the chords of the
circle C that subtend an angle of 21 at its center is
[2006]
(a) x² + y² = 3
© x² + y2 = 27
(6) x² + y² =1
(a) x² + y2 =
11
302The lines ( 2 x-3 y=5 ) and ( 3 x-4 y=7 )
are the diameters of a circle of area 154
square units. An equation of this circle is ( (boldsymbol{pi}=mathbf{2 2} / mathbf{7}) )
A ( cdot x^{2}+y^{2}+2 x-2 y=62 )
B. ( x^{2}+y^{2}+2 x-2 y=47 )
c. ( x^{2}+y^{2}-2 x+2 y=47 )
D. ( x^{2}+y^{2}-2 x+2 y=62 )
11
303A circle with center (3,8) contains the point ( (2,-1) . ) Another point on the circle is:
в. (4,17)
c. (5,-9)
D. (7,15)
E . (9,6)
11
304The equations of the tangents drawn from the origin to the
circle x2 + y2 – 2rx – 2hy + h2 = 0, are (1988 – 2 Marks)
(a) x=0
(b) y=0
(c) (h2 – r2)x – 2rhy=0 (d) (h2 – p2)x+ 2rhy=0
11
305If the eccentricity of an ellipse is ( frac{5}{8} ) and
the distance between its foci is ( 10, ) then
find latus-rectum of the ellipse.
11
306Eccentricity of the conic represented by ( boldsymbol{x}=frac{boldsymbol{e}^{boldsymbol{a}}+boldsymbol{e}^{-boldsymbol{a}}}{mathbf{2} sqrt{mathbf{3}}}, boldsymbol{y}=frac{boldsymbol{e}^{boldsymbol{a}}-boldsymbol{e}^{-boldsymbol{a}}}{mathbf{2}} ) is11
307Find the equation of the circle whose two end points of the diameter are
(4,-2) and (-1,3)
11
308Find the latus rectum of the parabola ( boldsymbol{x}^{2}+mathbf{2} boldsymbol{y}-mathbf{3} boldsymbol{x}+mathbf{5}=mathbf{0} )
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D.
11
309Find the equation of the circle with centre (1,1) and radius ( sqrt{2} )11
310A circle of radius 5 units touches the
coordinate axes in the first quadrant. If the circle makes one complete roll on ( x ) axis along the positive direction of ( x ) axis, find its equation in new position.
11
311The sum of the focal distances of a
point on the ellipse ( frac{x^{2}}{4}+frac{y^{2}}{9}=1 ) is:
A . 4 units
B. 6 units
c. 8 units
D. 10 units
11
312The magnitude of the gradient of the tangent at an extremity of latera recta of the hyperbola ( frac{x^{2}}{a^{2}}-frac{y^{2}}{b^{2}}=1 ) is equal
to (where ( e ) is the eccentricity of the hyperbola)
A ( . ) be
в.
( c cdot a b )
D. ( a )
11
313The eccentricity of the conic ( frac{mathbf{7}}{r}=mathbf{2}+ )
( 3 cos theta ) is
A . -3
в. 3
( c cdot frac{3}{2} )
D.
11
314The length of the latusrectum of an ellipse is equal to one-half of its minor axis. Then the eccentricity of the ellipse is :
A ( cdot frac{sqrt{3}}{2} )
B. ( frac{sqrt{2}}{3} )
c. ( frac{1}{sqrt{3}} )
D. ( frac{1}{sqrt{2}} )
11
315What are represented by the equation ( boldsymbol{x}^{3}+boldsymbol{y}^{3}+(boldsymbol{x}+boldsymbol{y})(boldsymbol{x} boldsymbol{y}-boldsymbol{a} boldsymbol{x}-boldsymbol{a} boldsymbol{y})=mathbf{0} )11
316For the conic ( 9 x^{2}-16 y^{2}+18 x+ )
( 32 y-151=0, ) find ( e )
A ( cdot frac{5}{4} )
в. ( frac{4}{5} )
( c cdot frac{1}{2} )
D. None of these
11
317The eccentricity of the conic ( 9 x^{2}+ )
( 25 y^{2}=225 ) is
A ( cdot frac{2}{5} )
B. ( frac{4}{5} )
( c cdot frac{1}{3} )
D.
E.
11
318Prove that the eccentricity of the conic given by the general equation satisfies the relation ( frac{e^{4}}{1-e^{2}}+4= )
( frac{(a+b-2 h cos omega)^{2}}{left(a b-h^{2}right) sin ^{2} omega}, ) where ( omega ) is the
angle between the axes.
11
319If the normal at the end of latus rectum of the ellipse ( frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1 ) passes
through ( (0,-b), ) then ( e^{4}+e^{2} ) (where ( E ) is eccentricity) equals
A cdot eccentricity of the ellipse is ( sqrt{frac{sqrt{5}-1}{2}} )
B. ratio of the minor and major axes is ( frac{sqrt{5}+1}{2} )
c. square of the eccentricity is equal to the ratio of the minor and major axes
D. none of these
11
3205.
The abscissa of the two points A and B are the roots of the
equation x2 + 2ax – b2 = 0 and their ordinates are the roots
of the equation x2 + 2px – q2 = 0. Find the equation and the
radius of the circle with AB as diameter. (1984 – 4 Marks)
11
32110. A circle touches the line y = x at a point P such that
OP= 412, where is the origin. The circle contains the
point (-10, 2) in its interior and the length of its chord on
the line x + y = 0 is 62. Determine the equation of the
circle.
(1990 – 5 Marks)
11
322Find the equation to the ellipses, whose centres are the origin, whose axes are the axes of coordinates, and which pass through ( (alpha) ) the points ( (2,2), ) and (3,1) and ( (beta) ) the points (1,4) and (-6,1)11
323Find the equation of the ellipse referred to its centre whose foci are the points
(4,0) and (4,0) and whose eccentricity is ( frac{1}{3} )
11
324Find the equation of the circle passing through the points (0,-1) and (2,0) and whose centre lies on the line ( 3 x+ )
( boldsymbol{y}=mathbf{5} )
11
325Find the equation of the circle which pass through the origin and cut off intercepts ( a ) and ( b ) respectively from the
( x ) and ( y ) axes.
11
326Find the value of ( a ) for which the ellipse ( frac{x^{2}}{y^{2}}+frac{y^{2}}{b^{2}}=1,(a>b), ) if the extremities
of the latus rectum of the ellipse having positive ordinates lie on the parabola ( x^{2}=-2(y-2) )
11
327If the eccentricity of an ellipse is ( frac{5}{8} ) and
the distance between its foci is ( 10, ) then
find the latusrectum of the ellipse.
11
328If the latus rectum of an ellipse is equal to half of minor axis, then find its
eccentricity.
11
32928. Consider a branch of the hyperbola
x2 – 2y2 – 2V3x – 4/2y—6= 0
with vertex at the point A. Let B be one of the end points of
its latus rectum. If C is the focus of the hyperbola nearest to
the point A, then the area of the triangle ABC is (2008)
11
330The length of the latus rectum of the parabola ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}^{2}+boldsymbol{b} boldsymbol{y}+boldsymbol{c} ) is ( frac{boldsymbol{k}}{boldsymbol{a}} . ) Find ( boldsymbol{k} )11
331An ellipse, with foci at (0,2) and (0,-2) and mirror axis of length ( 4, ) passes through which of the following points?
A ( cdot(1,2 sqrt{2}) )
B. ( (2, sqrt{2}) )
D. ( (sqrt{2}, 2) )
11
332Trace the following central conics. ( boldsymbol{x} boldsymbol{y}-boldsymbol{y}^{2}=boldsymbol{a}^{2} )11
333On the ellipse, ( 9 x^{2}+25 y^{2}=225, ) find
the point whose distance to the focus
( F_{1} ) is four times the distance to the
other focus ( boldsymbol{F}_{2} )
( mathbf{A} cdot[-15, sqrt{63}] )
( ^{text {В }} cdotleft(frac{-15}{4}, frac{sqrt{63}}{2}right) )
( ^{mathrm{c}} cdotleft(frac{-15}{4}, frac{sqrt{63}}{4}right) )
D. ( left(frac{-15}{2}, frac{sqrt{63}}{2}right) )
11
33429. The line passing through the extremity A of the major avis
and extremity B of the minor axis of the ellipse
x2 +9y2=9 .
meets its auxiliary circle at the point M. Then the area of the
triangle with vertices at A, Mand the origin Ois (2009)
31
ZI
(b) 10
() 10
(d) 27
11
335The equation of the ellipse whose foci ( operatorname{are}(pm 5,0) ) and of the directrix is ( 5 x= )
( 36, ) is
A ( frac{x^{2}}{36}+frac{y^{2}}{11}=1 )
B. ( frac{x^{2}}{6}+frac{y^{2}}{sqrt{11}}=1 )
c. ( frac{x^{2}}{6}+frac{y^{2}}{11}=1 )
D. None of these
11
336Find the equation of normal to the parabola ( y^{2}=4 a x ) which passes
through the point ( (-6 a, 0) ). and
suspended at ( 60^{circ} )
11
337Find the co-ordinates of the foci, the
vertices, the length of major axis, latus rectum and the eccentricity of the conic
represented by the equation ( 3 x^{2}+ )
( 5 y^{2}=15 )
11
338An ellipse has ( O B ) as semi-minor axis, ( F ) and ( F^{prime} ) its foci and the ( angle F B F^{prime} ) is a
right angle. Then, the eccentricity of the ellipse is
A ( cdot frac{1}{sqrt{3}} )
B. ( frac{1}{4} )
( c cdot frac{1}{2} )
D. ( frac{1}{sqrt{2}} )
11
339Find ( a ) and ( b ) for the ellipse ( b^{2} x^{2}+ ) ( a^{2} y^{2}=a^{2} b^{2} ) if the distance between the
directrices is ( 5 sqrt{5} ) and the distance
between the foci is ( 4 sqrt{5} ). Give answer to
the nearest integer
11
340If the eccentricity of a hyperbola is ( frac{5}{4} ) then find the eccentricity of its conjugate hyperbola.11
3415.
The abscissa of the two points A and B are the roots of the
equation x2 + 2ax – b2 = 0 and their ordinates are the roots
of the equation x2 + 2px – q2 = 0. Find the equation and the
radius of the circle with AB as diameter. (1984 – 4 Marks)
11
342Find the equation to the parabola whose focus is (1,-1) and vertex is (2,1)11
343Find the equation of the circle which
passes through (2,3) and (4,5) and the centre lies on the straight line ( y- )
( 4 x+3=0 )
11
344A parabola with axis parallel to ( x ) axis
passes through ( (mathbf{0}, mathbf{0}),(mathbf{2}, mathbf{1}),(mathbf{4},-mathbf{1}) . ) Its
length of latus rectum is
A ( cdot frac{2}{3} )
B.
( c cdot frac{7}{3} )
D. ( frac{1}{3} )
11
345The foci of the hyperbola ( 16 x^{2}-9 y^{2}- )
( 64 x+18 y-90=0 ) are
( ^{mathrm{A}} cdotleft(frac{24 pm 5 sqrt{145}}{12}, 1right) )
B. ( left(frac{21 pm 5 sqrt{145}}{12}, 1right) )
c. ( left(1, frac{24 pm 5 sqrt{145}}{2}right) )
D ( cdotleft(1, frac{21 pm 5 sqrt{145}}{2}right) )
E ( cdotleft(frac{21 pm 5 sqrt{145}}{2},-1right) )
11
346Find the equation of the circle with
center at (-3,5) and passes through
the point (5,-1)
A ( cdot(x+3)^{2}+(y-5)^{2}=100 )
B. ( (x-3)^{2}+(y-5)^{2} )
C. ( (x+3)^{2}+(y-5)^{2} )
D. None of the above
11
347Find the equation of parabola whose
focus is ( S(1,-7) ) and vertex is
( boldsymbol{A}(mathbf{1},-mathbf{2}) )
11
348The length of latus rectum of the parabola ( (x-2 a)^{2}+y^{2}=x^{2} ) is
A ( .2 a )
B. ( 3 a )
( c cdot 6 a )
D. ( 4 a )
11
349Find the equation of the circle circumscribing a square ABCD with side I and ( A B ) and ( A D ) as coordinate axes11
350The equation of the image of the circle ( x^{2}+y^{2}-6 x-4 y+12=0 ) by the
line mirror ( boldsymbol{x}+boldsymbol{y}-mathbf{1}=mathbf{0} ) is
A ( cdot x^{2}+y^{2}+2 x+4 y+4=0 )
B . ( x^{2}+y^{2}-2 x+4 y+4=0 )
c. ( x^{2}+y^{2}+2 x+4 y-4=0 )
D. ( x^{2}+y^{2}+2 x-4 y+4=0 )
11
351The point (3,4) is the focus and ( 2 x- )
( 3 y+5=0 ) is the directrix of a
parabola. Lenghth of latus rectum is
A ( cdot frac{2}{sqrt{13}} )
в. ( frac{4}{sqrt{13}} )
c. ( frac{1}{sqrt{13}} )
D. ( frac{3}{sqrt{13}} )
11
352The equation of ellipse whose major axis is along the direction of ( x ) -axis, eccentricity is ( e=2 / 3 )
A ( cdot 36 x^{2}+20 y^{2}=405 )
B . ( 20 x^{2}+36 y^{2}=405 )
c. ( 30 x^{2}+22 y^{2}=411 )
D. ( 22 x^{2}+32 y^{2}=409 )
11
353The focus of an ellipse is at the origin. The directrix is the line ( x=4 ) and its
eccentricity is ( frac{1}{2} ) then length of its semi major axis is
A ( cdot frac{2}{3} )
B. ( frac{4}{3} )
( c cdot frac{5}{3} )
D. ( frac{8}{3} )
11
35458
The tangent PT and the normal PNT
at a point P on it meet its axis at
The locus of the centroid of the trian
whose
ormal PN to the parabola y2 =4ax
S axis at points T and N, respectively.
d of the triangle PTN is a parabola
(2009)
(a)
vertex is
(b) directrix is x=0
(©) latus rectum is a
(d) focus is (a,0)
11
355The parabola ( (boldsymbol{y}+mathbf{1})^{2}=boldsymbol{a}(boldsymbol{x}-boldsymbol{2}) )
passes through the point (1,-2) then the equation of its directrix is
A ( .4 x+1=0 )
B. ( 4 x-1=0 )
( mathbf{c} cdot 4 x+9=0 )
D. ( 4 x-9=0 )
11
35610. Let AB be a chord of the circle x2 + y2 = r2 subtending a right
angle at the centre. Then the locus of the centroid of the
triangle PAB as P moves on the circle is
(20015)
(a) a parabola
(b) a circle
(c) an ellipse
(d) a pair of straight lines
11
3571.
The number of values of c such that the straight line
y= 4x+c touches the curve (x2/4) + y2 = 1 is
(1998 – 2 Marks)
(a) 0 (b) 1 (c) 2 (d) infinite.
11
358If ( mathbf{S} ) and ( mathbf{S}^{prime} ) are the foci and ( mathbf{B} ) is an
endpoint of the minor axis of an ellipse. If ( angle S B S^{prime}=120^{circ} ) then its eccentricity
is:
A ( cdot frac{sqrt{5}}{2} )
B. ( frac{sqrt{3}}{2} )
c. ( frac{1}{sqrt{2}} )
D. ( frac{1}{sqrt{3}} )
11
359The ellipse ( frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1 ) passes
through the point (-3,1) and has the eccentricity ( sqrt{frac{2}{5}} . ) Then the major axis of the ellipse has the length:
( ^{A} cdot 4 sqrt{frac{2}{5}} )
B. ( 8 sqrt{frac{2}{3}} )
( ^{c} cdot 4 sqrt{frac{2}{3}} )
D. ( 8 sqrt{frac{2}{5}} )
11
360The equation of directrix and latus
rectum of a parabola are ( 3 x-4 y+ )
( 27=0 ) and ( 3 x-4 y+2=0 . ) Then the
length of latus rectum is
( mathbf{A} cdot mathbf{5} )
B . 10
c. 15
D. 20
11
361Length of the latus rectum of the parabola ( sqrt{boldsymbol{x}}+sqrt{boldsymbol{y}}=sqrt{boldsymbol{a}} ) is
begin{tabular}{l}
A ( cdot a sqrt{2} ) \
hline
end{tabular}
B. ( frac{a}{sqrt{2}} )
( c )
D. 2a
11
362Equation of the ellipse in its standard form is ( frac{boldsymbol{x}^{2}}{boldsymbol{a}^{2}}-frac{boldsymbol{y}^{2}}{boldsymbol{b}^{2}}=mathbf{1} )
A. True
B. False
c. Nither
D. Either
11
363(1
)
Ро
17. Let PQ and RS be tangents at the extremities of the diameter
PR of a circle of radius r. If PS and RQ intersect at a point X
on the circumference of the circle, then 2r equals (20015)
(a)
PO.RS
(b) (PQ+RS)/2
(C) 2PQ.RS/(PQ+RS)
(d)
PO2 + RS2)/2
o
11
3642.
A is a point on the parabola y2 = 4ax. The normal at A cuts
the parabola again at point B. If AB subtends a right angle at
the vertex of the parabola. find the slope of AB.
(1982-5 Marks)
11
365DU DULULU
PO
20. If P and Q are the points of
then there is a circle passing throu
the points of intersection of the circles
y + 3x + 7y + 2p-5= 0 and x2 + y2 + 2x +2y-p2=0
circle passing through P, Q and (1, 1) for:
[2009]
(a) all except one value of p
(b) all except two values of p
(C) exactly one value of p
(d) all values of p
The cirol. 2
21.
11
36625. The slope of the line touching both the parabolas y2 = 4x
and x2 = -32y is
[JEE M 2014]
11
367An ellipse intersects the hyperbola 2×2 – 2y2 = 1 orthogonally.
The eccentricity of the ellipse is reciprocal of that of the
hyperbola. If the axes of the ellipse are along the coordinate
axes, then
(2009)
(a) equation of ellipse is x2 + 2y2 = 2
(b) the foci of ellipse are (-1,0)
(C) equation of ellipse is x2 + 2y2 = 4
(d) the foci of ellipse are(V2,0)
11
368Find the coordinates of the focus,
equation of the directrix and the latus
rectum of
( 2 x^{2}+3 y=0 )
11
369Find the vertex focus, equation of directrix and equation of axis of the ( boldsymbol{y}^{2}-boldsymbol{x}+mathbf{4} boldsymbol{y}+mathbf{5}=mathbf{0} )11
370P2P 4
Circle(s) touching x-axis at a distance 3 from the origin and
having an intercept of length 277 on y-axis is (are)
(JEE Adv. 2013)
(a) x2 + y2 – 6x + 8y+9=0 (b) x2 + y2 – 6x + 7y+9=0
(c) x2 + y2 – 6x – 8y+9=0 (d) x2 + y2 – 6x – 7y+9=0
11
371U 35<m<85
The two circles x2 + y = ax
each other if
(a) lal=0
(c) Ja=2c
(d) -85 0) touch
[2011]
(b) a=20
(d) 2a =C
ha 41
11
37211.
If the circles x2 + y2
+ 2ax + cy + a = 0 and
for
x2 + y2 – 3 ax + dy-1=0 intersect in two distinct points P
and Q then the line 5x + by – a=0 passes through P and Q
[2005]
(a) exactly one value of a
(b) no value of a
(c) infinitely many values of a
(d) exactly two values of a
maunhan the circle with
11
373If the eccentricity of the ellipse ( a x^{2}+ )
( 4 y^{2}=4 a,(a<4) ) is ( frac{1}{sqrt{2}}, ) then its semi minor axis is equal to
A .2
B. ( sqrt{2} )
c. 1
D. ( sqrt{3} )
E . 3
11
37440.
Ifthe line y=mx + 7.3 is normal to the hyperbol
24 18
= 1, then a value of m is : JEE M 2019-9 April (M)
yperbola
11
375If a 20 and the line 2bx + 3cy +4d = 0 passes th
the points of intersection of the parabolas
y2 = 4ax and x2 = 4ay, then
[2004]
(a) d2 + (36 – 2c)2 = 0 (b) d2 + (3b +2c)2 = 0
() d? +(26 – 3c)2 = 0 (d) d? +(26+ 3c)2 = 0
11
376The foci of a hyperbola coincide with the foci of the ellipse ( frac{x^{2}}{25}+frac{y^{2}}{9}=1 . ) Then
the equation hyperbola with eccentricity 2 is
A ( frac{x^{2}}{12}-frac{y^{2}}{4}=1 )
B. ( frac{x^{2}}{4}-frac{y^{2}}{12}=1 )
c. ( 3 x^{2}-y^{2}+12=0 )
D. ( 9 x^{2}-25 y^{2}-225=0 )
11
377Define Eccentricity?11
378The point (3,4) is the focus and ( 2 x- )
( 3 y+5=0 ) is the directrix of a parabola
Its latus rectum is
A ( cdot frac{2}{sqrt{13}} )
в. ( frac{4}{sqrt{13}} )
c. ( frac{1}{sqrt{13}} )
D. ( frac{3}{sqrt{13}} )
11
379If a hyperbola passes through the foci of the ellipse ( frac{x^{2}}{25}+frac{y^{2}}{16}=1 ) and its
traverse and conjugate axis coincide
with major and minor axes of the
ellipse, and product of the
eccentricities is 1, then:
This question has multiple correct options
A eqaations of the hyperbola is ( frac{x^{2}}{9}-frac{y^{2}}{16}=1 )
B – Equations of the hyperbola is ( frac{x^{2}}{9}-frac{y^{2}}{25}=1 )
C. Focus of the hyperbola is (5,0)
D. Focus of the hyperbola is ( (5 sqrt{3}, 0) )
11
380Through the vertex O of parabola y2 = 4x, chords OP and
OQ are drawn at right angles to one another. Show that for
all positions of P, PQ cuts the axis of the parabola at a fixed
point. Also find the locus of the middle point of PQ.
(1994 – 4 Marks)
11
381
POOD
(2011)
12. Let L be a normal to the parabola y2 = 4x. If LP
El passes througe
the point (9,6), then L is given by
(a) y-x+3=0
(c) y + x-15 = 0
(d) y-2x + 12 = 0
(b) y + 3x – 33 = 0
11
382A circle passes through the point (2,1) and the line ( x+2 y=1 ) is a tangent to it at the point (3,-1) Determine its equation.11
3836.
The radius of the circle passing through the foci of the
ellipse
-= 1, and having its centre at (0,3) is
lo x 22
(1995S)
(a) 4
(b) 3
©
2
11
384vertices of an ellipse are (0,±10) and its eccentricity ( e=4 / 5 ) then its
equation is
A ( cdot 90 x^{2}-40 y^{2}=3600 )
B . ( 80 x^{2}+50 y^{2}=4000 )
c. ( 36 x^{2}+100 y^{2}=3600 )
D. ( 100 x^{2}+36 y^{2}=3600 )
11
385What does the equation ( frac{x^{2}}{r-1}- ) ( frac{y^{2}}{1+r}=1, r>1, ) represents11
386The eccentricity of the conic represented by ( sqrt{(x+2)^{2}+y^{2}}+ )
( sqrt{(x-2)^{2}+y^{2}}=8 ) is
A ( cdot frac{1}{3} )
в. ( frac{1}{2} )
( c cdot frac{1}{4} )
D.
11
387If the eccentricity of the hyperbola ( x^{2}- ) ( y^{2} sec ^{2} alpha=5 ) is ( sqrt{3} ) times the
eccentricity of the ellipse ( boldsymbol{x}^{2} mathbf{s e c}^{2} boldsymbol{alpha}+ )
( boldsymbol{y}^{2}=25, ) then the value of ( boldsymbol{alpha} ) is
( mathbf{A} cdot pi / 6 )
в. ( pi / 4 )
c. ( pi / 3 )
D. ( pi / 2 )
11
388The equation ( 2 x^{2}+3 y^{2}-8 x-18 y+ )
( mathbf{3 5}=boldsymbol{lambda} ) represents?
A . A circle for all ( lambda )
B. An ellipse if ( lambda0 )
D. A-point if ( lambda=0 )
11
38910.
the line y = x at a point P such that
ere O is the origin. The circle contains the
A circle touches the line y = x at a pou
OP = 472 , where O is the origin. The circle
point (- 10, 2) in its interior and the length of its chord on
the line x + y = 0 is 62. Determine the equation of the
circle.
(1990 – 5 Marks)
11
39027. The re
27.
The number of common tan
sualgul le Palallel l y un
number of common tangents to the circles x4 + y2
– 6x – 12=0 and x2 + y2 +6x +18y+26=0, is :
[JEE M 2015)
(a) 3 (6) 4 (0) 1 (d) 2
into the circles x2 + y2-4;
11
391Find the radius and centre of the circle
( x^{2}+y^{2}-24 y+128=0 )
11
392For hyperbola ( frac{x^{2}}{cos ^{2} a}-frac{y^{2}}{sin ^{2} a}=1 )
which of the following remains constant with change in ‘a’?
A. Abscissae of vertices
B. Abscissae of foci
c. Eccentricity
D. Directrix
11
393Let ( a, b ) be non-zero real numbers. The
equation
( left(a x^{2}+b y^{2}+cright)left(x^{2}-5 x y+6 y^{2}right) )
represents
A. four straight lines, when ( c=0 ) and ( a, b ) are of the same ( operatorname{sign} )
B. two straight lines and a circle, when ( a=b ) and ( c ) is of sign opposite to that of ( a )
C. two straight lines and a hyperbola, when ( a ) and ( b ) are of the same sign and ( c ) is of sign opposite to that of ( a )
D. a circle and an ellipse, when ( a ) and ( b ) are of the same
( operatorname{sign} )
11
3948.
A variable circle passes through the fixed point A(p,q) and
touches x-axis. The locus of the other end of the diameter
through A is
[2004]
(a) (y-q)2 = 4 px (b) (x – 9)2 = 4 py
(c) (y-p)2 = 4qx (d) (x − p)2 = 4qy
11
395Find the coordinates of the foci, the
vertices the eccentricity and the length
of latus rectum of the hyperbola ( 16 x^{2}- )
( 9 y^{2}=576 )
11
396If latus rectum of ellipse ( frac{x^{2}}{25}+frac{y^{2}}{16}=1 )
is double ordiante passing through
focus of parabola ( boldsymbol{y}^{2}=mathbf{4} boldsymbol{a} boldsymbol{x}, ) then find
the value of ( a ).
11
397An ellipse passing through origin has its foci at (5,12) and ( (24,7), ) then its eccentricity is
A ( cdot frac{sqrt{386}}{38} )
B. ( frac{sqrt{386}}{39} )
c. ( frac{sqrt{386}}{47} )
D. ( frac{sqrt{386}}{51} )
11
398If ( e_{1} ) and ( e_{2} ) are the eccentricities of two
conics with ( e_{1}^{2}+e_{2}^{2}=3, ) then the conics
are
A. Ellipses
B. Parabolas
c. circles
D. Hyperbolas
11
399Find the coordinates of focus, the
equation of the directrix and length of the latus rectum of the conic
represented by the equation ( x^{2}=-y )
11
400The equations of the common tangents to the parabola
y=x2 and y=-(x – 2)2 is/are
(2006 – 5M, -1)
(a) y=4 (x-1)
(b) y=0
(c) y=4(x – 1)
(d) y=-30x – 50
11
40115.
Find the intervals of values
um um Pum).
e intervals of values of a for which the line y+x=0
bisects two chords drawn from a point —
(1+V2a 1-2a
32
to the circle 2x² + 2y2 – (1+V2a)x -(1-V2a)y=0).
(100
ml
11
402( A ) and ( B ) are fixed points. If ( |P A-P B| )
is a constant, locus of ( boldsymbol{P} ) is
A. a parabola
B. an ellipse
c. a hyperbola
D. a circle
11
4031.
A square is inscribed in the circle x2 + y2 – 2x + 4y+3=0. Its
sides are parallel to the coordinate axes. The one vertex of
the square is
(1980)
(a) (1+ V2,-2) (b) (1– V2,-2)
(C) (1, -2+ 2) (d) none of these
11
404In a parabola, length of the latus rectum
is 4a.
A . True
B. False
11
405Normal of the parabola ( x^{2}=8 y ) at (2,1)
is
A. ( 2 x+y+5=0 )
в. ( 2 x+y=5 )
( mathbf{c} cdot 2 x-y+5=0 )
D. ( 2 x-y-5=0 )
11
406Find the equation of the ellipse whose
vertices are ( (pm mathbf{7}, mathbf{0}) ) and foci are
( (pm mathbf{6}, mathbf{0}) )
11
40738. Three circles of radii a, b, co
I radii a, b, c (a <b< c) touch each other
ley have x-axis as a common tangent, then:
JJEEM 2019-9 Jan (
MI
(c)
(d)
do Jo Jo
a, b, c are in A.P
Vā, Vī, Vc are in A.P.
11
4088.
An ellipse has OB as semi minor axis, F and Fits focii and
the angle FBF’ is a right angle. Then the eccentricity of
the ellipse is
[2005]
11
409Find the equation of the ellipse whose
vertices are (±8,0) and foci are
( (pm mathbf{3}, mathbf{0}) )
11
41018.
C, and C, are two concentric circles, the radius of C, being
twice that of C. From a point P on C,, tangents PA and PB
are drawn to C. Prove that the centroid of the triangle PAB
(1998 -8 Marks)
lies on C,
11
411The ellipse ( frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1 ) cuts ( x ) axis at
A and y axis at ( mathrm{B} ) and the line joining the focus ( mathrm{S} ) and ( mathrm{B} ) makes an angle ( frac{3 pi}{4} ) with x-axis. Then the eccentricity of the ellipse is
A. ( frac{1}{sqrt{2}} )
в. ( frac{1}{2} )
c. ( frac{sqrt{3}}{2} )
D. ( frac{1}{3} )
11
41213.
If two distinct chords, drawn from the point (p, q) on the
circle x2 + y2 = px + qy (where pq 0) are bisected by the
x-axis, then
(1999 – 2 Marks)
(a) p2 = 22 (b) p2 = 892 (c) P2 8q2.
11
413Find the equation of directrix and
length of latus rectum of the parabola ( x^{2}=16 y )
A. ( y-4=0 ) and 16
B. ( y+4=0 ) and 16
c. ( y-4=0 ) and 4
D. ( y+4=0 ) and 4
11
414Find the area bounded by the ellipse ( frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1 ) and the ordinates ( x=0 )
and ( x=a e, ) where ( b^{2}=a^{2}left(1-e^{2}right) ) and
( e<1 )
11
415+2
a
12. For the Hyperbola –
-=1, which of the
cos’ a sind a
following remains constant when a varies = ? (2007)
(a) abscissae of vertices (b) abscissae of foci
(c) eccentricity
(d) directrix.
11
416If the curve ( y=|x-3| ) touches the parabola ( boldsymbol{y}^{2}=boldsymbol{lambda}(boldsymbol{x}-mathbf{4}), boldsymbol{lambda}>mathbf{0}, ) then
latus rectum of the parabola, is
( A cdot 2 )
B. 4
( c cdot 8 )
( D cdot 16 )
11
41768. Find the coordinates of the cen-
tre of the circle passing through
the points (0, 0), (-2, 1) and (-3,
21.
(3) (11,3)
(4) (3, 11)
11
41811.
Angle between the tanger
=x2 – 5x + 6
te between the tangents to the curve y = x
[2006]
at the points (2,0) and (3,0) is
(a) T
(b)
()
(d)
11
419The area of the region bounded by the curve ( sqrt{boldsymbol{x}}+sqrt{boldsymbol{y}}=sqrt{boldsymbol{a}}(boldsymbol{x}, boldsymbol{y} geq 0) ) and the
coordinate axes is
A ( cdot a^{2} / 6 )
в. ( a^{2} / 2 )
c. ( a^{2} / 3 )
D. ( a^{2} )
11
420The equation ( frac{x^{2}}{2-r}+frac{y^{2}}{r-5}+1=0 )
represents an ellipse, if
( mathbf{A} cdot r>2 )
В. ( 2<r5 )
D. ( r in(2,5) )
11
421Find the coordinates of focus, the
equation of the directrix and length of the latus rectum of the conic
represented by the equation ( 3 x^{2}=-y )
11
422Find the equation to the ellipse, whose focus is the point ( (-1,1), ) whose directrix is the straight line ( boldsymbol{x}-boldsymbol{y}+ ) ( mathbf{3}=mathbf{0}, ) and whose eccentricity is ( frac{mathbf{1}}{mathbf{2}} )11
423If the distance of one of the focus of
hyperbola from the two directrices of hyperbola are 5 and ( 3, ) then its eccentricity is
A ( cdot sqrt{2} )
B. 2
( c cdot 4 )
D.
11
424Eccentricity of the hyperbola whose
asymptotes are given by ( 3 x+2 y+ )
( mathbf{5}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}+mathbf{3} boldsymbol{y}+mathbf{5}=mathbf{0} ) is
A ( cdot sqrt{2} )
B. ( frac{3}{2} )
( c cdot 2 )
D. None of these
11
425An ellipse passes through the point (4,-1) and touches the line ( x+4 y- )
( 10=0 . ) Find its equation if its axes
coincide with the coordinate axes.
11
426On the parabola ( y=x^{2}, ) the point least
distant from the straight line ( y=2 x- )
4 is
( mathbf{A} cdot(1,1) )
B ( cdot(1,0) )
c. (1,-1)
D ( cdot(0,0) )
11
427If the major axis of an ellipse is three times the minor axis, then its
eccentricity is equal to
A ( cdot frac{1}{3} )
B. ( frac{1}{sqrt{3}} )
c. ( frac{1}{sqrt{2}} )
D. ( frac{2 sqrt{2}}{3} )
E ( frac{2}{3 sqrt{2}} )
11
428Find the vertex, axis, focus,
directrix,lastusrectum of the parabola. ( x^{2}-2 x+4 y+9=0 )
11
429Let circles ( C_{1} ) and ( C_{2} ) an Argand plane
be given by ( |z+1|=3 ) and ( |z-2|=7 )
respectively. If a variable circle
( left|z-z_{0}right|=r quad ) be inside circle ( C_{2} ) such
that it touches ( C_{1} ) externally and ( C_{2} )
internally then locus of ( z_{0} ) describes a
conic ( boldsymbol{E} ) whose eccentricity is equal to
A ( cdot frac{1}{10} )
B. ( frac{3}{10} )
c. ( frac{5}{10} )
D. ( frac{7}{10} )
11
430Find the area of a quadrant of a circle whose circumference is ( 44 mathrm{cm} )11
431The eccentricity of the hyperbola whose latus-return is 8 and length of the conjugate axis is equal to half the distance between the foci, is
A ( cdot frac{4}{3} )
B. ( frac{4}{sqrt{3}} )
c. ( frac{2}{sqrt{3}} )
D. None of these
11
432Axis of a parabola is ( y=x ) and vertex
and focus are at a distance ( sqrt{2} ) and ( 2 sqrt{2} ) respectively from the origin. The equation of the parabola is
A ( cdot(x-y)^{2}=8(x+y-2) )
B . ( (x+y)^{2}=2(x+y-2) )
c. ( (x-y)^{2}=4(x+y-2) )
D. ( (x+y)^{2}=2(x-y+2) )
11
433The Vertex of the parabola ( boldsymbol{y}^{2}-mathbf{1 0} boldsymbol{y}+ )
( boldsymbol{x}+mathbf{2 2}=mathbf{0} ) is.
( A cdot(3,4) )
B. (3,5)
( c cdot(5,3) )
D. none of these
11
434Ratio of the greatest and least focal distances of a point on the ellipse ( 4 x^{2}+9 y^{2}=36 ) is:
A. ( 4+sqrt{5}: 4-sqrt{5} )
B. ( 5+sqrt{5}: 4-sqrt{3} )
c. ( 3+sqrt{5}: 3-sqrt{5} )
D. ( sqrt{7}: 2 )
11
435Length of the latusrectum of the
hyperbola ( boldsymbol{x} boldsymbol{y}-boldsymbol{3} boldsymbol{x}-boldsymbol{4} boldsymbol{y}+boldsymbol{8}=boldsymbol{0} ) is
A .4
B. ( 4 sqrt{2} )
( c cdot 8 )
D. None of these
11
436Let A be the centre of the circle x2 + y2 – 2x – 4y – 20 = 0.
Suppose that the tangents at the points
B(1,7) and D(4.-2) on the circle meet at the point C. Find
the area of the quadrilateral ABCD. (1981 – 4 Marks)
11
437Consider the conic ( e x^{2}+pi y^{2}-2 e^{2} x- )
( 2 pi^{2} y+e^{3}+pi^{3}=pi e . ) Suppose ( P ) is any
point on the conic and ( S_{1}, S_{2} ) are the
foci of the conic, then the maximum
value of ( left(boldsymbol{P S}_{1}+boldsymbol{P S}_{2}right) ) is
A ( . pi e )
в. ( sqrt{pi e} )
( c cdot 2 sqrt{pi} )
D. ( 2 sqrt{e} )
11
438fe is the eccentricity of the ellipse ( frac{x^{2}}{16}+frac{y^{2}}{25}=1 ) and ( e_{2} ) is the eccentricity
of the hyperbola passing through the foci of the ellipse and ( e_{1} e_{2}=1, ) then equation of the hyperbola is
( stackrel{mathbf{A}}{*} frac{x^{2}}{9}-frac{y^{2}}{16}=1 )
в. ( frac{x^{2}}{16}-frac{y^{2}}{9}=-1 )
c. ( frac{x^{2}}{9}-frac{y^{2}}{25}=1 )
D. ( frac{x^{2}}{25}-frac{y^{2}}{9}=1 )
11
439Find the coordinates of the foci, the
vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse ( frac{x^{2}}{25}+frac{y^{2}}{100}=1 )
11
440In the ellipse distance between the foci
is equal to the distance between a focus
and one end of minor axis then its
eccentricity is
A ( cdot frac{1}{2} )
B.
( c cdot frac{1}{3} )
D.
11
4412.
Let A be the centre of the circle x2 + y2 – 2x – 4y – 20 = 0.
Suppose that the tangents at the points
B(1,7) and D(4.-2) on the circle meet at the point C. Find
the area of the quadrilateral ABCD. (1981 – 4 Marks)
11
442The co-ordinates of the focus of
parabola ( boldsymbol{y}^{2}=mathbf{2 0} boldsymbol{x} ) are
A. (5,5)
(年) 5,5,5
B. (3,5)
c. (0,5)
(年. (0,5)
D. (5,0)
11
443The locus of the mid-point of a chord of the circle
x + y2 = 4 which subtends a right angle at the origin 15
as a right angle at the origin is
(1984 – 2 Marks)
(a) x+y=2
(b) x2 + y2 = 1
(c) x² + y2=2
(d) x+y=1
11
444A straight line and a point not lying on it
are given on a plane. Find the set of
points which are equidistant from the given straight line and the given point.
11
445The equation ( 7 y^{2}-9 x^{2}+54 x-28 y- )
( 116=0 ) represents
A. a hyperbola
B. a parabola
c. an ellipse
D. a pair of straight lines
11
446If the tangent to the curve, ( y=x^{3}+ )
( a x-b ) at the point (1,-5) is
perpendicular to the line, ( -boldsymbol{x}+boldsymbol{y}+ )
( 4=0, ) then which one of the following
points lies on the curve?
A. (-2,2)
)
в. (2,-2)
c. (2,-1)
D. (-2,1)
11
447Consider the hyperbola
center N(X2,0). Suppose
point P(x,y) with x;
er the hyperbola H:r2 – 12 = 1 and a circle S with
2,0). Suppose that Hand S touch each other at a
Hand Sat
1:1) with r, >I and y, >0. The common tangent to
and Sat Pintersects the x-axis at point M. If(L, m) is the
roid of the triangle PMN, then the correct expression(s)
(JEE Adv. 2015
is(are)
(a) Pop-1-bank for x; >1
dm
(d) dvi – 1 for x > 0
TL
11
448بح
13.
Tangents are drawn to the hyperbola
= 1, parallel
to the straight line 2x – y = 1. The points of conta
tangents on the hyperbola are
• The points of contact of the
(2012)
(C) (313,- 2/2)
(d) (-313,212)
11
449The curve represented by ( boldsymbol{R} boldsymbol{s}left(frac{1}{z}right)=boldsymbol{C} )
is (where ( C ) is a constant and ( neq 0 ) )
A. Ellipse
B. Parabola
c. circle
D. Straight line
11
45023.
The circle passing through the point (-1,0) and touching
the y-axis at (0, 2) also passes through the point. (2011)
@ (0)6 (2) © C
a (-4,0)
1.defontent
11
45172
Pola
19. For hyperbola –
-=1 which of the following
cos- a sin a
remains constant with change in ‘a’
(2003)
(a) abscissae of vertices (b) abscissae of foci
(c) eccentricity
(d) directrix
11
452If the ( operatorname{lines} 3 x-4 y-7=0 ) and ( 2 x- )
( 3 y-5=0 ) are two diameters of a circle
of area 154 square units, the equation of the circle is :
A. ( x^{2}+y^{2}+2 x-2 y-62=0 )
B . ( x^{2}+y^{2}-2 x+2 y-62=0 )
c. ( x^{2}+y^{2}-2 x+2 y-47=0 )
D. ( x^{2}+y^{2}+2 x-2 y-47=0 )
11
45320. If tangents are drawn to the ellipse x2 + 2y2 = 2, then the
locus of the mid-point of the intercept made by the tangents
between the coordinate axes is
(2004S)
(b
2x
11
454Trace the following central conics. ( boldsymbol{x} boldsymbol{y}=boldsymbol{a}(boldsymbol{x}+boldsymbol{y}) )11
455Find the Lactus Rectum of ( 9 y^{2}- )
( 4 x^{2}=36 )
( mathbf{A} cdot mathbf{9} )
B. 6
c. 11
D. 15
11
45619. Equation of the ellipse whose axes are the axes of
coordinates and which passes through the point (-3,1) and
has eccentricity v
(a) 5×2 + 3y2 – 48=0
(c) 5x² +37² – 32=0
[2011]
(b) 3×2 + 5y2 –15=0
(d) 3x² +577 – 32=0
11
457The length of latus rectum of the ellipse ( 4 x^{2}+9 y^{2}=36 ) is
A ( cdot frac{4}{3} )
B. ( frac{8}{3} )
( c .6 )
D. 12
11
458U2C
(
U2UU
The length of the diameter of the circle which to
x-axis at the point (1,0) and passes through the p
[2012]
TV
11
4593.
Find the equations of the circle passing through (4,3) and
touching the lines x+y= 2 and x – y = 2. (1982 – 3 Marks)
1..
.
11
460Assume that water issuing from the end of a horizontal pipe, 7.5 m above the ground, describes a parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position ( 2.5 mathrm{m} ) below the line of the pipe, the flow of water has curved outward from ( 3 mathrm{m} ), beyond the
vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground?
11
461The equation ( frac{x^{2}}{10-lambda}+frac{y^{2}}{6-lambda}=1 )
represents This question has multiple correct options
A. a hyperbola if ( lambda6 )
c. a hyperbola if ( 6<lambda<10 )
D. an ellipse if ( 0<lambda<6 )
11
462The equation of the ellipse having
vertices at (±5,0) and foci (±4,0) is
( ^{A} cdot frac{x^{2}}{25}+frac{y^{2}}{16}=1 )
B. ( 9 x^{2}+25 y^{2}=225 )
c. ( frac{x^{2}}{9}+frac{y^{2}}{25}=1 )
D. ( 4 x^{2}+5 y^{2}=20 )
11
463Find the equation to the conic passing through the origin and the points
( (1,1),(-1,1),(2,0), ) and (3,-2)
Determine its species.
11
464In what ratio, the point of intersection of the common tangents to hyperbola ( frac{x^{2}}{1}-frac{y^{2}}{8}=1 ) and parabola ( y^{2}=12 x )
divides the foci of the given hyperbola?
A . 3: 4
B. 3: 2
( c .5: 4 )
D. 5: 3
11
465Trace the following central conics. ( boldsymbol{x}^{2}+boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}+boldsymbol{x}+boldsymbol{y}+mathbf{1}=mathbf{0} )11
466The D on of family of circles with radius ( =5 & ) and center on ( y=2 ) is11
467Center of the hyperbola ( x^{2}+ ) ( 4 y^{2}+6 x y+8 x-2 y+7=0 quad ) is
A ( .(1,1) )
в. (0,2)
( c cdot(2,0) )
D. None of these
11
468Find the eccentricity, foci and the length
of the latus-rectum of the ellipse ( x^{2}+ )
( 4 y^{2}+8 y-2 x+1=0 )
11
469The eccentricity of the ellipse ( 12 x^{2}+ ) ( 7 y^{2}=84 ) is equal to:
A ( frac{sqrt{5}}{7} )
B. ( sqrt{frac{5}{7}} )
c. ( frac{sqrt{5}}{12} )
D.
E. ( frac{7}{12} )
11
470If one of the diameters of the circle, given by the equation,
x2 + y2 – 4x + 6y-12=0, is a chord of a circle S, whose centre
is at (3, 2), then the radius of Sis:
JJEEM 2016]
(a) 5
(b) 10
(c) 512
(d) 513
11
4713.
Three normals are drawn from the point (c, 0) to the curve
y=x. Show that c must be greater than 1/2. One normal is
always the x-axis. Find c for which the other two normals are
perpendicular to each other.
(1991 – 4 Marks)
T
.
11_D1
11
472Find the coordinates of the focus axis of
the parabola the equation of directrix
and the length of the latus rectum for ( y^{2} )
( =-8 x )
11
473Assertion
The equation ( 3 x^{2}-2 y^{2}+4 x-6 y=0 )
represents a hyperbola.
Reason
The second degree equation ( a x^{2}+ ) ( 2 h x y+b y^{2}+2 g x+2 f y+c=0 )
represents a hyperbola if ( a b c+2 f g h- )
( boldsymbol{a} boldsymbol{f}^{2}-boldsymbol{b} boldsymbol{g}^{2}-boldsymbol{c h}^{2} neq boldsymbol{0} & boldsymbol{h}^{2}>boldsymbol{a b} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
11
474If the line ( x-1=0 ) is the directrix of
the parabola ( y^{2}-k x+8=0, ) then one
of the values of ( k ) is
A ( .1 / 8 )
B. 8
( c cdot 4 )
D. ( 1 / 4 )
11
47522. Tangents drawn from the point P(1,8) to the circle
x2 + y2-6x-4-11=0
touch the circle at the points A and B. The equation of the
circumcircle of the triangle PAB is
(2009)
(a) x2 + y2 + 4x-6y +19=0
(b) x2 + y2 – 4x – 10y +19=0
(C) x2 + y2 – 2x + 6y – 29=0
(d) x2 + y2 – 6x – 4y +19=0
11
476Let’d be the perpendicular distance from the centre of the
ellipse
Pse +2
72
62
-=1 to the tangent drawn at a point P on

the ellipse. If F, and Fy are the two foci of the ellipse, then
62)
show that (PF
(1995 – 5 Marks)
11
477In an ellipse the length of major axis is 10 and the distance between the foci is
8. Then the length of minor axis is:
A . 5
B. 7
( c cdot 4 )
D. 6
11
47830.
49-12=U
The eccentricity of the hyperbola whose leng
rectum is equal to 8 and the length of its con
equal to half of the distance between its foci,
erbola whose length of the latus
the length of its conjugate axis is
[JEE M 2016]
4
11
479The curve ( 5 x^{2}+12 x y-22 x-12 y- )
( mathbf{1 9}=mathbf{0} ) is.
A . Ellipse
B. Parabola
c. Hyperbola
D. Parallel straight lines
11
480Find the equation of the circle with
centre (-1,-2) and radius 5
11
481The equation of the circumcircle of the triangle formed by the lines ( y+sqrt{3} x= ) ( mathbf{6}, boldsymbol{y}-sqrt{mathbf{3}} boldsymbol{x}=mathbf{6} ) and ( boldsymbol{y}=mathbf{0} ) is
A ( cdot x^{2}+y^{2}+4 x=0 )
В. ( x^{2}+y^{2}-4 y=0 )
c. ( x^{2}+y^{2}-4 y=12 )
D. ( x^{2}+y^{2}+4 x=12 )
11
482TO
Tf2x -y+1=0 is a tangent to the hyperbola –
the hyperbola 22 161,
then which of the following cannot be sides of a right angled
triangle?
(JEE Adv. 2017)
(a) a, 4,1
(b) a, 4,2
(c) 2a, 8, 1
(d) 2a, 4,1
11
483The eccentricity of an ellipse, with its centre a
ellipse, with its centre at the origin, is
If one of the directrices is x =
I the directrices is x = 4, then the equation of
the ellipse is:
[2004]
(a) 4x² +3y2 = 1
(b) 3×2 + 4y2 = 12
©) 4x² +3y2 = 12
(d) 3×2 + 4y2 = 1
11
484Prove that the locus of the point of intersection of the lines
( x cos alpha+y sin alpha=a )
and ( x sin alpha-y cos alpha=b )
is a circle whatever ( alpha ) may be.
11
485Fo parabola ( 3 y^{2}=16 x, ) equation of
directrix and length of latus rectum is
A ( cdot x=frac{-4}{3}, frac{3}{4} )
B. ( x=frac{4}{3}, frac{4}{3} )
c. ( y=frac{-4}{3}, frac{4}{3} )
D. ( x=frac{-4}{3}, frac{16}{3} )
11
48610. One of the diameters of the circle circumscribing
the rectangle ABCD is 4y = x+7. If A and B are the
points (-3, 4) and (5, 4) respectively, then find the area of
rectangle.
(1985 – 3 Marks)
11
487Find the equation of the hyperbola satisfying the give conditions: Foci
(0,±13) the conjugate axis is of length
( mathbf{2 4} )
11
488Find the equation of the ellipse whose
foci are (0,±7) and length of the minor
axis is 26
11
489A focus of an ellipse is at the origin. The directrix is the line ( mathbf{x}=mathbf{4} ) and the eccentricity is ( frac{1}{2} . ) Then the length of the semi major axis is
A ( cdot frac{5}{3} )
B. ( frac{8}{3} )
( c cdot frac{2}{3} )
D. ( frac{4}{3} )
11
490Form the differential equation representing the family of ellipses
having centre at the origin and foci on ( x ) axis.
11
491The eccentricity of the conic ( r^{2} cos 2 theta=a^{2} ) is
( mathbf{A} cdot mathbf{1} )
B. ( sqrt{2} )
( c cdot 2 )
D.
11
492The foci of an ellipse are ( S(-2,-3), S^{prime}(0,1) ) and its ( e=frac{1}{sqrt{2}} )
then the directrix corresponding to the
focus ( S^{prime} ) is:
A. ( x+2 y-5=0 )
B . ( x+2 y-9=0 )
c. ( x+2 y-11=0 )
D. none of these
11
493to the chord y = mx + 1 of the circle x-+y+=1 subtends an
angle of measure 45° at the major segment of the circle then
value of mis
[2002]
(a) 2012 (b) -2
(C) -1972
none of these
11
49433. Let (x, y) be any point on the parabola y< =4x. Let P be the
point that divides the line segment from (0,0) to (x, y) in the
ratio 1: 3. Then the locus of Pis
(2011)
(a) x2=y (b) y2 = 2x (c) y2 = x (d) x2 = 2y
11
495Find the equation of the hyperbola satisfying the given conditions: Foci ( (pm 3 sqrt{5}, 0) ) the latus rectum is of length 811
496If foci are points (0,1)(0,-1) and minor axis is of length ( 1, ) then equation of ellipse is
A ( cdot frac{x^{2}}{1 / 4}+frac{y^{2}}{5 / 4}=1 )
В. ( frac{x^{2}}{5 / 4}+frac{y^{2}}{1 / 4}=1 )
c. ( frac{x^{2}}{3 / 4}+frac{y^{2}}{1 / 4}=1 )
D. ( frac{x^{2}}{1 / 4}+frac{y^{2}}{3 / 4}=1 )
11
497JV 2
If a tangent to the circle x2 + y2 = lintersects the coordinate
axes at distinct points P and Q, then the locus of the
mid-point of PQ is:
(JEEM 2019-9 April (M)]
(a) x2 + y2 – 4x2y2=0 (b) x2 + y2– 2xy=0
(©) x2 + y2 – 16xży2=0 (d) x2 + y2– 2x2y2=0
11
498The focus of the parabola ( boldsymbol{y}^{2}-boldsymbol{x}- )
( 2 y+2=0 ) is
( ^{mathrm{A}} cdotleft(frac{1}{4}, 0right) )
в. (1,2)
c. ( left(frac{5}{4}, 1right) )
D. ( left(frac{3}{4}, frac{5}{2}right) )
11
499At some point ( mathbf{P} ) on the ellipse, the segment ( mathbf{S S}^{1} ) subtends a right angle,
then its eccentricity is
( ^{A} cdot_{e}=frac{sqrt{2}}{2} )
в. ( efrac{1}{sqrt{2}} )
D. ( frac{sqrt{3}}{2} )
11
500Equation ( (2+lambda) x^{2}-2 lambda x y+(lambda- )
1) ( y^{2}-4 x-2=0 ) represents a
hyperbola if
A ( . lambda=4 )
B . ( lambda=1 )
( c cdot lambda=frac{4}{3} )
( D cdot lambda=3 )
11
50132. The radius of a circle, having minimum area, which touches
the curve y=4 – x2 and the lines, y= |x|is:
(JEEM 2018]
(a) 4(+2+1) (b) 2(V2 +1)
(c) 2(√2-1) (d) 41/2-1).
11
502If the latus rectum of an ellipse is equal to half of minor axis, find its
eccentricity.
11
503The line ( (x-2) cos theta+(y-2) sin theta= )
1 touches a circle for all value of ( theta ), then
the equation of circle is
A ( cdot x^{2}+y^{2}-4 x-4 y+7=0 )
B . ( x^{2}+y^{2}+4 x+4 y+7=0 )
c. ( x^{2}+y^{2}-4 x-4 y-7=0 )
D. None of the above
11
504Assertion(A): The difference of the focal
distances of any point on the hyperbola ( frac{boldsymbol{x}^{2}}{mathbf{3 6}}-frac{boldsymbol{y}^{2}}{mathbf{9}}=mathbf{1} ) is 12
Reason(R): The difference of the focal
distances of any point on the hyperbola is equal to the length of it transverse axis
A. Both A and R are true and R is the correct
explanation of ( A )
B. Both A and R are true but R is not the correct
explanation of ( A )
C. A is true but R is false.
D. A is false but R is true.
11
505The equation ( x^{2}+9=2 y^{2} ) is an
example of which of the following
curves?
A. hyperbola
B. circle
c. ellipse
D. parabola
E. line
11
506Find the coordinates of the focus,
equation of the directrix and the latus rectum of
( boldsymbol{x}^{2}=-2 boldsymbol{y} )
11
507The equation ( frac{x^{2}}{2-a}+frac{y^{2}}{a-5}+1=0 )
represents an ellipse if ( boldsymbol{a} in )
( ^{mathbf{A}} cdotleft(2, frac{3}{2}right) cupleft(frac{3}{2}, 5right) )
в. ( left(2, frac{3}{2}right) )
c. ( left(1, frac{3}{2}right) )
D. ( left(frac{3}{2}, 5right) )
11
508Let ( C ) be the circle with centre at (1,1)
and radius ( =1 . ) If ( T ) is the circle
centered at ( (0, y), ) passing through origin and touching the circle ( C )
externally, then the radius of ( T ) is equal
to :
A ( cdot frac{sqrt{3}}{sqrt{2}} )
B. ( frac{sqrt{3}}{2} )
( c cdot frac{1}{2} )
D.
11
509Find the locus of a point which moves so that the difference of its distances from
the points, (5,0) and (-5,0) is 2 is:
A ( cdot frac{x^{2}}{1}+frac{y^{2}}{24}=1 )
B. ( frac{x^{2}}{24}+frac{y^{2}}{1}=1 )
c. ( frac{x^{2}}{24}-frac{y^{2}}{2}=1 )
D. ( frac{x^{2}}{1}-frac{y^{2}}{24}=1 )
11
510Assertion ( (A): ) The eccentricity of an ellipse is ( frac{mathbf{3}}{mathbf{5}} )
Reason ( (boldsymbol{R}): ) The equation of the ellipse is ( x=5 cos theta, y=4 sin theta )
A. Assertion and reason both are correct and reason is an explanation.
B. Assertion and reason both are correct and reason is not an explanation
c. Assertion is incorrect and reason is correct
D. Assertion is correct and reason is incorrect
11
51113.
The equation of the directrix of the parabola
y2 + 4y + 4x + 2 =0 is
(a) x=-1 (b) x=1 (c) x=-3/2 (d)
(20015)
x=3/2
for which
11
512Find the equation of tangents to the hyperbola ( 3 x^{2}-4 y^{2}=12, ) which make
equal intercepts on the axes.
11
513Arrange the eccentricities of the following ellipses in particular order.
A- Ellipse whose Latus rectum is half of its major axis.
B- Ellipse whose distance between the foci is equal to the distance between a focus and one end of minor axis
C- Ellipse whose major axis is double the minor axis.
D- Ellipse whose distance between the foci is 6 and the length of minor axis is
8
( A )
[
(A)=(C)>(D)>(B)
]
B. ( (A)=(C)<(D)(C)>(D)>(B)
]
D. none of these
11
514From the following information, find the equation of Hyperbola and the equation of its Transverse Axis:
Focus ( :(-2,1), ) Directrix ( : 2 x-3 y+ ) ( mathbf{1}, boldsymbol{e}=frac{mathbf{2}}{sqrt{mathbf{3}}} )
11
51521. An ellipse is drawn by taking a diameter of the circle (x-1)-
+ y2 = 1 as its semi-minor axis and a diameter of the circle
x +(y-2)2 =4 is semi-major axis. If the centre of the ellipse
is at the origin and its axes are the coordinate axes, then the
equation of the ellipse is:
[2012]
(a) 4×2 + y2 =4
(b) x2 + 4y2 = 8
(c) 4×2 + y2 = 8
(d) x2 + 4y2 = 16
e
1
11
516Find the locus of the point of intersection of the lines ( sqrt{3} x-y- ) ( 4 sqrt{3} lambda=0 ) and ( sqrt{3} lambda x+lambda y-4 sqrt{3}=0 )
for different values of ( lambda )
A ( cdot 4 x^{2}-y^{2}=48 )
B . ( x^{2}-4 y^{2}=48 )
c. ( 3 x^{2}-y^{2}=48 )
D. ( y^{2}-3 x^{2}=48 )
11
517The ends of major axis of an ellipse are (5,0)(-5,0) and one of the foci lies on ( 3 x-5 y-9=0, ) then the eccentricity
of the ellipse is
A ( cdot frac{2}{3} )
B. ( frac{3}{5} )
( c cdot frac{4}{5} )
D.
11
518The equation of the hyperbola whose foci are the foci of the ellipse ( frac{x^{2}}{25}+ ) ( frac{boldsymbol{y}^{2}}{mathbf{9}}=1 ) and the eccentricity is ( mathbf{2}, ) is11
5193.
The centre of the circle passing through the point (O
touching the curve y = x2 at (2.4) is (1983 – 1 Mark)
ssing through the point (0, 1) and
-16
(6)
(-16 53)
(d) none of these
11
520The circles ( boldsymbol{x}^{2}+boldsymbol{y}^{2}-boldsymbol{4} boldsymbol{x}+boldsymbol{4} boldsymbol{y}+boldsymbol{4}=mathbf{0} )
and ( x^{2}+y^{2}-4 x-4 y=0 )
A. Do not intersect
B. Are not orthogonal
c. Intersect orthogonally
D. concentric
11
52118.
If the tangent at the point Pon the circle x + y + 6x+6y=2
meets a straight line 5x-2y+6=0 at a point Q on the y-axis,
then the length of PQ is
(2002)
(a) 4 (6) 2/5 () 5 (d) 3/5
11
522If the equation ( (10 x-5)^{2}+(10 y- )
4)( ^{2}=lambda^{2}(3 x+4 y-1)^{2} ) represents a
hyperbola, then :
A ( .-2<lambda2 )
c. ( lambda2 )
11
52335. The common tangents to the circle x2 + y2 = 2 and the
parabola y2 = 8x touch the circle at the points P, Q and the
parabola at the points R, S. Then the area of the quadrilateral
PQRS is
(JEE Adv. 2014)
(a) 3 (b) 6 40 (c) 9 (d) 15
11
524Find the equation of the ellipse whose
foci are (0,±3) and length of the minor
axis is 16
11
52520.
If one of the diameters of the circle x2 + y2 – 2x – 6y+6=0
is a chord to the circle with centre (2, 1), then the radius of
the circle is
(2004S)
a) v3 (b) VI (C) 3 (d) 2
11
5269.
If | m;, -i,m.>0.1=1. 2. 3. 4 are four distinct points on a
circle, then show that my m2m2m4 = 1
(1989 – 2 Marks)
how that m m mzma = 1
11
527Circles are drawn on chords of the
rectangular hyperbola ( boldsymbol{x} boldsymbol{y}=mathbf{4} ) parallel to the line ( y=x ) as diameters.All such
circles pass through two fixed points whose coordinates are
This question has multiple correct options
( mathbf{A} cdot(2,2) )
B ( cdot(2,-2) )
c. (-2,2)
D. (-2,-2)
11
5284
A point on the parabola y2 = 18x at which the ordinate
increases at twice the rate of the abscissa is [2004]
(a)
6) (2-4)
(a) (2,4)
11
529Find the equation of the circle of minimum radius which contains the
three circles
( boldsymbol{S}_{1} equiv boldsymbol{x}^{2}+boldsymbol{y}^{2}-boldsymbol{4} boldsymbol{y}-boldsymbol{5}=boldsymbol{0} )
( boldsymbol{S}_{2} equiv boldsymbol{x}^{2}+boldsymbol{y}^{2}+mathbf{1 2} boldsymbol{x}+boldsymbol{4} boldsymbol{y}+boldsymbol{3 1}=mathbf{0} )
and ( S_{2} equiv x^{2}+y^{2}+6 x+12 y+ )
( mathbf{3 6}=mathbf{0} )
11
530Find the area bounded by the ellipse ( frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1 ) and the ordinates ( x=0 )
and ( x=a e, ) where ( b^{2}=a^{2}left(1-e^{2}right) ) and
( e<1 )
11
531The equation of the circle passing through (4,5) having the centre at (2,2) is
A ( cdot x^{2}+y^{2}+4 x+4 y-5=0 )
B . ( x^{2}+y^{2}-4 x-4 y-5=0 )
c. ( x^{2}+y^{2}-4 x=13 )
D. ( x^{2}+y^{2}-4 x-4 y+5=0 )
11
53211.
Two circles, each of radius 5 units, touch each other at (
the equation of their common tangent is 4x +3y=10, find
the equation of the circles.
(1991 – 4 Marks)
11
533The hyperbola ( frac{x^{2}}{a^{2}}-frac{y^{2}}{b^{2}}=1 ) passes
through the point ( (sqrt{6}, 3) ) and the length
of the latusrectum is ( frac{18}{5} . ) Then, the
length of the transverse axis is equal to
A . 5
B. 4
( c cdot 3 )
D. 2
E . 1
11
534Arrange the ellipses in ascending order of their eccentricities when ( angle S B S^{1} ) is
given, where ( S & S^{1} ) are foci and ( B ) is
one end of the minor axis. For the four
ellipses ( boldsymbol{A}, boldsymbol{B}, boldsymbol{C}, boldsymbol{D} angle mathbf{S B S}^{1}= )
( mathbf{2 0}^{mathbf{0}}, mathbf{6 0}^{mathbf{0}}, mathbf{3 0}^{mathbf{0}}, mathbf{9 0}^{mathbf{0}} )
( A cdot A, C, B, D )
B. D, B, C, A
c. ( B, D, C, A )
D. ( A, C, D, B )
11
535The radius of the circle ( x^{2}+y^{2}-5 x+ )
( 2 y+5=0 ) is
( A cdot 2 )
B.
( c cdot frac{3}{2} )
D. ( frac{2}{3} )
11
536Length of the latus rectum of the
hyperbola ( boldsymbol{x} boldsymbol{y}-boldsymbol{3} boldsymbol{x}-boldsymbol{4} boldsymbol{y}+boldsymbol{8}=mathbf{0} )
A .4
B. ( 4 sqrt{2} )
c. 8
D. none of these
11
537If the equation ( frac{lambda(x+1)^{2}}{3}+ ) ( frac{(y+2)^{2}}{4}=1 ) represents a circle then
( lambda= )
( mathbf{A} cdot mathbf{1} )
в. ( frac{3}{4} )
c. 0
D. ( -frac{3}{4} )
11
538Find the equation of circle touching the line 2x + 3y +1=0
at (1, -1) and cutting orthogonally the circle having line
segment joining (0, 3) and (-2,-1) as diameter.
am
11
539For parabola ( boldsymbol{y}^{2}=mathbf{8 4} boldsymbol{x}, ) focal distance of
point (21,1764) is
A . 64
B. 84
( c cdot 24 )
D. 42
11
54024. The locus of the foot of perpendicular drawn from the centre
of the ellipse x2 + 3y2 = 6 on any tangent to it is
(JEE M 2014]
(a) (x2 + y2 )° = 6x² +2y2 (b) (x2 + y2) * = 6×2 – 2y?
(©) (x² – y2)° = 6×2 + 2y2 (d) (x2 – y2) ° = 6×2 – 2y2
11
54110. Intercept on the line y = x by the circle x2 + y2 – 2x = 0 is
AB. Equation of the circle on AB as a diameter is [2004]
(a) x2 + y2 + x – y = 0
(b) x2 + y2 – x+y=0
(C) x² + y2 +x+y=0
(d) x2 + y2 – x – y = 0
11
542The triangle PQR is inscribed in the circle ( x^{2}+y^{2}=25 . ) If ( Q ) and ( R ) have
coordinates ( (mathbf{3}, mathbf{4}) ) and ( (-mathbf{4}, mathbf{3}) )
respectively, then ( angle Q P R ) is equal to
A ( cdot frac{pi}{2} )
в.
c.
D.
11
54330.
The normal at a point P on the ellipse x2 +4y2 = 16 mesta
x-axis at Q. If Mis the mid point of the line segment PO
then the locus of M intersects the latus rectums of the
given ellipse at the points
(2009)
3
(6 (+213, + 3 )
( [+2/3, + 4/3]
11
544What is the area enclosed by ( |x|+ ) ( |boldsymbol{y}|=mathbf{1} ? )
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D. 4
11
545Eccentricity of ellipse ( frac{x^{2}}{a^{2}+1}+frac{y^{2}}{a^{2}+2}= )
1 is ( frac{1}{sqrt{3}} ) then length of Latus rectum is
A ( cdot frac{2}{sqrt{3}} )
B. ( frac{4}{sqrt{3}} )
( c cdot 2 sqrt{3} )
D. ( frac{sqrt{3}}{2} )
11
546The locus of a point which moves such that the difference of its distances from
two fixed points is always a constant is
A . A straight line
B. A circle
c. An ellipse
D. A hyperbola
11
547Eccentricity of a hyperbola is always less than 1.
A. True
B. False
11
5486.
Lines 5x+12y-10=0 and 5x-12y -40=0 touch a circle C,
of diameter 6. If the centre of C, lies in the first quadrant,
find the equation of the circle C, which is concentric with
C, and cuts intercepts of length 8 on these lines.
11
549If ( t ) is a parameter, then ( x= ) ( left(t+frac{1}{t}right) y=bleft(t-frac{1}{t}right) ) represents
A. an ellipse
B. a circle
c. a pair of straight lines
D. a hyperbola
11
550From any point on the hyperbola ( frac{x^{2}}{a^{2}}- ) ( frac{y^{2}}{b^{2}}=1 ) tangents are drawn to the hyperbola ( frac{x^{2}}{a^{2}}-frac{y^{2}}{b^{2}}=2 . ) The area cut-off
by the chord of contact on the
asymptotes is equal to
A ( cdot frac{a b}{2} )
B. ab
( c cdot 2 a b )
( D cdot 4 a b )
11
55118.
The focal chord to y2 = 16x is tangent to (x – 6) + y = 2,
then the possible values of the slope of this chord, are
(2003)
(a) {-1,1}
(b) {-2,2}
(c) {-2,-1/2}
(d) {2,-1/2}
11
552
3.
If the circle x2 + y2 = aintersects the hyperbola xy = c2 in
four points P(x,, y,), Q(x2,y»), R(xz,yz), S(xq, y2), then
(1998 – 2 Marks)
(a) x, +x,+x2+x2=0 (b) y,+y,+ y +y = 0
(c) x.x2x2x4 = c4 (d) yyyy4 = c*
11
553Let the eccentricity of the hyperbola ( frac{x^{2}}{a^{2}}-frac{y^{2}}{b^{2}}=1 ) be reciprocal to that of the
ellipse ( x^{2}+4 y^{2}=4 . ) If the hyperbola
passes through a focus of the ellipse, then
This question has multiple correct options
A the equation of the hyperbola is ( frac{x^{2}}{3}-frac{y^{2}}{2}=1 )
B. a focus of the hyperbola is (2,0)
c. the eccentricity of the hyperbola is ( sqrt{frac{5}{3}} )
D. the equation of the hyperbola is ( x^{2}-3 y^{2}=3 )
11
554Number of intersecting points of the
( operatorname{conic} 4 x^{2}+9 y^{2}=1 ) and ( 4 x^{2}+y^{2}=4 )
is
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot 3 )
D. 0 (zero)
11
555The area cut off by the parabola ( y^{2}= ) ( 4 a x ) and its latus rectum is………, if ( a=3 )11
556If the two circles (x – 1)2 +(y – 3)2 = r2 and
2 1,2 – 8x + 2y + 8 = 0) intersect in two distinct point,
[2003]
> 2 (b) 2 <r < 8 (c) r < 2 (d) r = 2.
then
11
557The point ( P ) on the parabola ( y^{2}=4 a x )
for which IPR PQI is maximum, where R ( (-a, 0), Q(0, a), ) is
A ( cdot(a, 2 a) )
в. ( (a,-2 a) )
c. ( (4 a, 4 a) )
D. ( (4 a,-4 a) )
11
558Points A, B and C lie on the parabola y2 = 4ax. The tangents
to the parabola at A, B and C, taken in pairs, intersect at
points P, Q and R. Determine the ratio of the areas of the
triangles ABC and PQR.
(1996 – 3 Marks)
11
11
559The equation of a circle with
through equilateral trian
(a) r2+y2=9a2
(c) x² + y2=4a²
ation of a circle with origin as a centre and passing
equilateral triangle whose median is of length 3a is
(b) x2+y=16a2
[2002]
(d) x2 + y2=a2
11
560A circle and a parabola intersect at four points ( left(x_{1}, y_{1}right),left(x_{2}, y_{2}right),left(x_{3}, y_{3}right) ) and
( left(x_{4}, y_{4}right) . ) Then ( y_{1}+y_{2}+y_{3}+y_{4} ) is equal
to
( mathbf{A} cdot mathbf{4} )
B. 3/2
( c cdot 2 )
D.
11
561The foci of an ellipse are located at the points (2,4) and ( (2,-2) . ) The points (4,2) lies on the ellipse. If ( a ) and ( b )
represent the lengths of the semi-major and semi-minor axes respectively, then
the value of ( (a b)^{2} ) is equal to
A. ( 68+22 sqrt{10} )
В. ( 6+22 sqrt{10} )
c. ( 26+10 sqrt{10} )
11
56218. Iftwo tangents drawn from a point P to the parabola y2 = 4x
are at right angles, then the locus of P is
[2010]
(a) 2x+1=0
(b) x=-1
(c) 2x-1=0
(d) x=1
to
Fanation of the
line
11
563State the following statement is True or
False

A parabolic arch has a height 18 meters and span 24 meters. Then the height of the arch at 8 meters from the centre of the span is equal to 10
A. True
B. False

11
564Find the equation of a circle of radius 5 whose centre lies on ( x- ) axis and
passes through the point (2,3)
11
56512.
sally Iwo values of a
A circle touches the x-axis and also touches the circle with
centre at (0,3 ) and radius 2. The locus of the centre of the
circle is
[2005]
(a) an ellipse
(b) a circle
(©) a hyperbola
a parabola
11
56622. For the circle x2 + y2 = r2, find the value of r for which the
area enclosed by the tangents drawn from the point P (6,8)
to the circle and the chord of contact is maximum.
(2003 – 2 Marks)
11
567The circle drawn on the minor axis as
diameter passes through the foci of the ellipse ( mathbf{S}=mathbf{0}, ) then its eccentricity ( mathbf{e}= )
( mathbf{A} cdot sin 18^{0} )
B. ( sin 30^{circ} )
c. ( cos 45^{circ} )
D. ( cos 30^{circ} )
11
568If the line ( l x+m y+n=0 ) touches the
parabola ( boldsymbol{y}^{2}=mathbf{4} boldsymbol{a} boldsymbol{x}, ) prove that ( boldsymbol{l} boldsymbol{n}= )
( boldsymbol{a} boldsymbol{m}^{2} )
11
569if the distance between the foci is equal
to the length of the latus-rectum. Find
the eccentricity of the ellipse.
A. ( frac{sqrt{5}-1}{2} )
B. ( frac{sqrt{5}+1}{2} )
c. ( frac{sqrt{5}-1}{4} )
D. None of these
11
570The centre of the hyperbola ( 9 x^{2}-36 x- )
( 16 y^{2}+96 y-252=0 ) is
A ( .(2,3) )
в. (-2,-3)
c. (-2,3)
D. none of these
11
571The latus rectum subtends a right angle at the centre of the ellipse then its eccentricity is
( A cdot 2 sin 18^{circ} )
B. ( 2 cos 18^{circ} )
( c cdot 2 sin 54^{circ} )
( D cdot 2 cos 54^{circ} )
11
572The eccentricity of an ellipse ( 9 x^{2}+ )
( 16 y^{2}=144 ) is
A ( frac{sqrt{3}}{5} )
B. ( frac{sqrt{5}}{3} )
c. ( frac{sqrt{7}}{4} )
D.
11
573( S ) and ( T ) are the foci of an ellipse and ( B )
is an end of the minor axis. If ( triangle boldsymbol{S T B} ) is
an equilateral triangle, the eccentricity of the ellipse is equals to:
A ( cdot frac{1}{4} )
B. ( frac{1}{3} )
( c cdot frac{1}{2} )
D. ( frac{2}{3} )
E. None of these
11
574Find the equation of the circles passing through the point (2,8) touching the line ( 4 x-3 y-24=0 ) and ( 4 x+3 y- )
( 42=0 ) and having ( x ) coordinate of the
centre of the circle less than or equal to
8
11
575If the parabola ( y^{2}=4 a x ) passes
through (3,2) then the length of latus
rectum is
A ( cdot frac{1}{3} )
B. ( frac{2}{3} )
c. 1
D.
11
576Prove that the points (7,-9) and (11,3)
lie on a circle with centre at the origin.
Also its equation.
11
577Find the range of ( c ) for which the line
( boldsymbol{y}=boldsymbol{m} boldsymbol{x}+boldsymbol{c} ) touches the parabola ( boldsymbol{y}^{2}= )
( 8(x+2) )
11
578A man running round a race course note
that the sum of the distances of two
flag posts from him is 8 meters. The
area of the path he encloses in square meters if the distance between the flag
posts is 4 is:
( mathbf{A} cdot 15 pi )
в. ( 12 pi )
c. ( 18 pi )
D. ( 8 sqrt{3} pi )
11
579Eccentricity of hyperbola ( frac{boldsymbol{x}^{2}}{boldsymbol{k}}-frac{boldsymbol{y}^{2}}{boldsymbol{k}}=mathbf{1} )
A ( cdot sqrt{1+k} )
B. ( sqrt{1-k} )
( c cdot sqrt{2} )
( D cdot 2 sqrt{2} )
11
580If the vertex ( =(2,0) ) and the
extremities of the latus rectum are
(3,2) and ( (3,-2), ) then the equation of the parabola is
A ( cdot y^{2}=2 x-4 )
B . ( x^{2}=4 y-8 )
c. ( y^{2}=4 x-8 )
D. None
11
581If the focal distance of an end of the
minor axis of any ellipse (referred to its axes as the axes of ( x ) and ( y )
respectively) is ( k ) and the distance between the foci is ( 2 h, ) then its equation
is:
A ( cdot frac{x^{2}}{k^{2}}+frac{y^{2}}{k^{2}+h^{2}}=1 )
в. ( frac{x^{2}}{k^{2}}+frac{y^{2}}{h^{2}-k^{2}}=1 )
c. ( frac{x^{2}}{k^{2}}+frac{y^{2}}{k^{2}-h^{2}}=1 )
D. ( frac{x^{2}}{k^{2}}+frac{y^{2}}{h^{2}}=1 )
11

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