Circles Questions

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

List of circles Questions

Question No Questions Class
1 Find the values of ( x ) and ( y ) in the figures
given below
9
2 The range of values of ( a ) such that the
angle ( theta ) between the pair of tangents drawn from ( (a, 0) ) to the circle ( x^{2}+ ) ( y^{2}=1 ) satisfies ( frac{pi}{2}<theta<pi ), is
A ( .(1,2) )
B. ( (1, sqrt{2}) )
c. ( (-sqrt{2},-1) )
(年) ( (-sqrt{2},-1) )
D. ( (-sqrt{2},-1) cup(1, sqrt{2}) )
10
3 If the length of the common chord of two intersecting equal circles be ( 6 mathrm{cm} ) and
if the radius of each circle be ( 5 mathrm{cm} ) then
the distance between the centers of the
circle is
A. ( 7 mathrm{cm} )
B. ( 8 mathrm{cm} )
( c .9 mathrm{cm} )
D. none
9
4 n given figure ( angle P Q R=100^{circ}, ) where
( P, Q ) and ( R ) are points on a circle with
centre ( O . ) Then, ( angle O P R ) is
( A cdot 20 )
B. 10
( c cdot 30^{circ} )
D. 40
9
5 The coordinate that the chord ( x cos alpha+ )
( boldsymbol{y} sin boldsymbol{alpha}-boldsymbol{p}=mathbf{0} ) of ( boldsymbol{x}^{2}+boldsymbol{y}^{2}-boldsymbol{a}^{2}=mathbf{0} ) may
subtend a right angle at the centre of the circle is?
A ( cdot a^{2}=2 p^{2} )
B ( cdot p^{2}=2 a^{2} )
c. ( a=2 p )
D. ( p=2 a )
9
6 What will be ( angle X O Y ) if arc ( A B= ) arc
( X Y ) and ( angle A O B=60^{circ} ? )
( A cdot 30^{circ} )
B. ( 45^{circ} )
( c cdot 50^{circ} )
D. 60
9
7 Circle O has diameters AB and CD
perpendicular to each other. AM is any
chord intersecting ( mathrm{CD} ) at ( mathrm{P.} ) Then ( A P . overline{A M} ) is equal to:
A. ( overline{A O} . overline{O B} )
в. ( overline{A O} . overline{A B} )
( c cdot overline{C P} cdot overline{C D} )
D. ( overline{C P} . overline{P D} )
ह. ( overline{c O} . overline{O P} )
9
8 62. ABCD is a cyclic trapezium
such that AD||BC, if ZABC
= 70°, then the value of ZBCD
is :
(1) 60 (2) 70°
(3) 40 (4) 80°
9
9 The slope of the tangent to the curve ( y=int_{0}^{x} frac{d t}{1+t^{3}} ) at the point where ( x=1 ) is.
A ( cdot frac{1}{4} )
B. ( frac{1}{3} )
( c cdot frac{1}{2} )
D. 1
10
10 The radius of a circle is given as ( 15 mathrm{cm} )
and chord AB subtends an angle of ( 131^{circ} )
at the centre ( C ) of the circle.Using trigonometry ,calculate:
(i) the length of ( A B )
(ii) the distance of ( A B ) from the centre ( C )
9
11 The locus of the centre of the circles
which touch both the circles ( x^{2}+y^{2}= )
( a^{2} ) and ( x^{2}+y^{2}=4 a x ) externally has the
equation:
A ( cdot 12(x-a)^{2}-4 y^{2}=3 a^{2} )
B ( cdot 9(x-a)^{2}-5 y^{2}=2 a^{2} )
C ( cdot 8 x^{2}-3(y-a)^{2}=9 a^{2} )
D. None of these
9
12 How many tangents can be drawn on the circle of radius ( 5 mathrm{cm} ) form a point lying outside the circle at distance ( 9 mathrm{cm} )
from the center
10
13 72. Two circles are of radii 7 cm and
2 cm their centres being 13cm
apart. Then the length of direct
common tangent to the circles
between the points of contact is
(1) 12 cm (2) 15 cm
(3) 10 cm (4) 5 cm
10
14 Find the values of ( x ) and ( y )
A . ( x=10.3, y=12.7 )
B . ( x=12.9, y=15.6 )
C. ( x=15.3, y=12.3 )
D. ( x=19.3, y=15.4 )
9
15 If ( 9.2 mathrm{cm} ) is the diameter of a circle then
its radius is
A ( .4 .1 mathrm{cm} )
в. ( 4.6 mathrm{cm} )
c. ( 4.8 mathrm{cm} )
D. ( 4.3 mathrm{cm} )
9
16 68. AB = 8 cm and CD = 6 cm are
two parallel chords on the
same side of the centre of a
circle. The distance between
them is 1 cm. The radius of the
circle is
(1) 5 cm (2) 4 cm
(3) 3 cm (4) 2 cm
9
17 If the angle between two tangents drawn from an external point ( P ) to a
circle of radius a and a center ( O ), is ( 60^{circ} )
then find the length of ( O P )
10
18 Chords ( M N ) and ( R S ) of a circle
intersect at ( boldsymbol{P} ) outside the circle If
( boldsymbol{P N}=mathbf{3} boldsymbol{c m}, boldsymbol{M} boldsymbol{N}=mathbf{5} boldsymbol{c m}, boldsymbol{P} boldsymbol{R}=boldsymbol{2} boldsymbol{c m} )
then the value of ( S R ) is equal to
( mathbf{A} cdot 5 mathrm{cm} )
B. ( 8 mathrm{cm} )
c. ( 15 mathrm{cm} )
D. ( 10 mathrm{cm} )
9
19 Two parallel chords ( A B ) and ( C D ) are 3.9
cm apart and lie on opposite sides of the centre of a circle. If ( A B=1.4 mathrm{cm} )
and ( C D=4 mathrm{cm}, ) find the radius of the
circle.
A. ( 3 mathrm{cm} )
B. ( 3.2 mathrm{cm} )
( c .2 .3 mathrm{cm} )
D. ( 2 mathrm{cm} )
9
20 A perpendicular at the end of the radius of a circle is
A. diameter
B. tangent
c. chord
D. anyline
10
21 The tangent drawn at the end point of two pependicular diameter of a circle. prove that ( mathrm{PQ} ) and ( mathrm{RS} ) are parallel 10
22 A straight line ( x=y+2 ) touches the
( operatorname{circle} 4left(x^{2}+y^{2}right)=r^{2} . ) The value of ( r ) is
A ( cdot sqrt{2} )
B. ( 2 sqrt{2} )
( c cdot 2 )
( D )
9
23 In the given figure, if 0 is the centre of a
circle, ( P Q ) is a chord and the tangent
( P R ) at ( P ) makes an angle of ( 50^{circ} ) with ( P Q )
find ( angle P O Q )
( A cdot 40 )
в. 80
( c cdot 100 )
D. 120
10
24 A point ( A ) is ( 26 c m ) away from the centre
of a circle and the length of tangent
drawn from ( A ) to the circle is 24 cm. Find
the radius of the circle.
( mathbf{A} cdot 10 mathrm{cm} )
B. ( 20 mathrm{cm} )
( mathbf{c} cdot 25 c m )
D. ( 15 mathrm{cm} )
10
25 n given figure triangle ( mathrm{ABCCCC} )
circumscribes the circle with center 0
and radius ( 2 mathrm{cm} )
Area of ( Delta A B C ) is ( 16 mathrm{cm}^{2} ). find ( mathrm{AB} )
( 5 mathrm{cm} )
( 6 c )
( 7 mathrm{cm} )
10
26 Angle inscribed in a semi-circle is
( mathbf{A} cdot pi / 2 )
в. ( pi / 3 )
c. ( pi / 4 )
D.
9
27 Determine the maximum number of
common tangents that can be drawn for each pair of circles shown.
10
28 Circle with centre 0 and radius 25 cms
has a chord ( A B ) of length of 14 cms in it.
Find the area of triangle AOB?
9
29 The points of intersection of the line ( 4 x-3 y-10=0 ) and the circle
( x^{2}+y^{2}-2 x+4 y-20=0 )
are………………..and.
This question has multiple correct options
A ( .(4,2) )
в. (-2,-6)
D. (-2,-4)
10
30 Length of the chord joining the points
( P(alpha) ) and ( Q(beta) ) on the circle ( x^{2}+y^{2}= )
( a^{2} ) is
A ( cdot cos left(frac{alpha-beta}{2}right) )
в. ( 2 a sin left(frac{alpha-beta}{2}right) )
c. ( 2 a tan left(frac{alpha-beta}{2}right) )
D. ( 2 a csc left(frac{alpha-beta}{2}right) )
9
31 In a circle with centre ( 0, O D perp ) chord ( A B )
If BC is the diameter, then
( mathbf{A} cdot A C=B C )
В ( . O D=B D )
c. ( A C=2 O D )
D. none of these
9
32 In a circle of radius ( 13 mathrm{cm}, P Q ) and ( R S )
are two parallel chords of length ( 24 mathrm{cm} ) and IOcm respectively. The chords are on the opposite sides of the centre. The distance between the chords
is?
A. ( 7 mathrm{cm} )
B. ( 17 mathrm{cm} )
( c cdot 26 c m )
D. ( 12 mathrm{cm} )
9
33 Suppose you are given a circle. Give a construction to find its centre. 9
34 ( O ) is the centre of the circle having
radius ( 5 mathrm{cm} . O M perp ) chord ( A B . ) If
( boldsymbol{O} boldsymbol{M}=mathbf{4} mathrm{cm}, ) then the length of the
chord ( A B ) is
A. ( 6 mathrm{cm} )
B. ( 5 mathrm{cm} )
( c cdot 8 mathrm{cm} )
D. ( 10 mathrm{cm} )
9
35 69. The distance between the cen-
tres of the two circles with radii
4 cm and 9 cm is 13 cm. The
length of the direct common tan-
gent (between two points of con-
tact) is
(1) 13 cm (2) 153 cm
(3) 12 cm (4) 18 cm
9
36 If ( P ) is a point on a circle with centre ( C ) then the line drawn through ( P ) and perpendicular to CP is the tangent to the circle at the point ( P )
A. True
B. False
c. Either
D. Neither
10
37 If the diameter of a circle decreases to
its ( frac{1}{4} ) then its radius decreases to
A ( cdot frac{1}{2} )
B. 4
( c cdot frac{1}{4} )
D.
9
38 The radius of a circle with centre 0 is 7
( mathrm{cm} . ) Two radii OA and ( mathrm{OB} ) are drwan at right angles to each other. Find the areas of minor and major segments.
9
39 Two chords ( A B ) and ( C D ) of lengths ( 5 mathrm{cm} )
and ( 11 mathrm{cm} ) respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between ( A B ) and ( C D ) is ( 6 mathrm{cm} )
find the radius of the circle.
9
40 If two parallel chords on the same side
of the centre of a circle are ( 6 mathrm{cm} ) and 8
( mathrm{cm}, ) and they are ( 1 mathrm{cm} ) apart, then the diameter of the circle will be
( mathbf{A} cdot 14 mathrm{cm} )
B. ( 10 mathrm{cm} )
c. ( 8 mathrm{cm} )
D. ( 5 mathrm{cm} )
9
41 Find the measure of arc ( C D ).
A ( cdot 105 )
B . ( 55^{circ} )
( c cdot 108 )
D. 75
9
42 Find the points of intersection of the
line ( x-y+2=0 ) and the circle ( 2 x^{2}+ )
( 2 y^{2}-29 x-19 y+56=0 . ) Also
determine the length of the chord intercepted.
9
43 The line ( 4 y-3 x+lambda=0 ) touches the
circle ( x^{2}+y^{2}-4 x-8 y-5=0 . ) The
value of ( lambda ) is
( mathbf{A} cdot 29 )
B . 10
c. -35
D. None of these
10
44 Length of a chord of a circle is ( 24 mathrm{cm} )
and its distance from the centre is 5
( mathrm{cm} . ) Find the diameter of the circle.
9
45 Which of the following is/ are correct?
This question has multiple correct options
A. A line segment with its endpoints lying on a circle is called a chord of the circle.
B. A line that intersects a circle at exactly one point is called a tangent to the circle.
C. Angle in a semi-circle is a right angle.
D. Lengths of the two tangents to a circle from an external point are equal
10
46 n figure, chords ( overline{P Q} ) and ( overline{R S} ) intersect
at ( mathrm{T} . ) If ( boldsymbol{m} angle boldsymbol{R}=mathbf{5 0}^{boldsymbol{o}} ) and ( boldsymbol{m} angle boldsymbol{P}=mathbf{4 6}^{boldsymbol{o}} )
the number of degrees in minor arc PR
is
( A cdot 84 )
B. 168
( c cdot 42 )
D. 130
E. cannot be determine
9
47 In a circle whose radius is ( 8 mathrm{cm}, ) a chord
is drawn at a point ( 3 mathrm{cm} ). from the centre of the circle. The chord is divided
into two segments by a point on it. If one segment of the chord is ( 9 mathrm{cm}, ) What is the length of the other segment?
9
48 A circle has two equal chords ( A B ) and
( A C, ) chord ( A D ) bisects ( B C ) in ( E . ) If ( A C=12 )
and ( A E=8 c m, ) then the measure of ( A D )
is ?
A ( .24 mathrm{cm} )
B. ( 18.5 mathrm{cm} )
c. ( 18 mathrm{cm} )
D. ( 19 mathrm{cm} )
9
49 If radii of two concentric circles are 4
( mathrm{cm} ) and ( 5 mathrm{cm}, ) then the length of each chord of one circle which is tangent to the circle is
A. ( 3 mathrm{cm} )
B. ( 6 mathrm{cm} )
( c .9 mathrm{cm} )
D. ( 1 mathrm{cm} )
9
50 Draw a circle and mark a point in its
interior.
9
51 f ( boldsymbol{m}(boldsymbol{a} boldsymbol{r} boldsymbol{c} boldsymbol{B} boldsymbol{C})=boldsymbol{8} boldsymbol{0}^{o}, ) find ( boldsymbol{m}(boldsymbol{a} boldsymbol{r} boldsymbol{c} boldsymbol{C} boldsymbol{D}) )
A ( cdot 40^{circ} )
B. ( 80^{circ} )
( c cdot 120^{circ} )
D. ( 140^{circ} )
9
52 Draw a circle and mark a radius. 9
53 Determine the length of the chord common to the circles ( x^{2}+y^{2}= )
( 64 a n d x^{2}+y^{2}-16 x=0 )
A ( cdot 2 sqrt{3} )
B. ( 4 sqrt{3} )
( c cdot 6 sqrt{3} )
D. ( 8 sqrt{3} )
9
54 74. ‘O’ is the circumcentre of trian-
gle ABC. If Z BAC = 50° then Z
OBC is
(1) 50°
(2) 100
(3) 130° (4) 40°
9
55 In the diagram, ( O ) is the centre of the
circle. Given that ( O Q= )
( 5 mathrm{cm} ) and ( A N=8 mathrm{cm}, ) find the length
of ( boldsymbol{P Q} )
9
56 Find the length of a chord that is at a distance of ( 5 mathrm{cm} ) form the centre of a
circle of radius ( 13 mathrm{cm} )
( A cdot 20 mathrm{cm} )
B . ( 24 mathrm{cm} )
c. ( 15 mathrm{cm} )
D. ( 12 mathrm{cm} )
9
57 Find the equation of tangent at (3,4) for the circle ( x^{2}+y^{2}=25 )
A. ( 3 x+4 y=25 )
в. ( 3 x-4 y=25 )
c. ( 3 x+4 y+25=0 )
D. ( 4 x-3 y=25 )
10
58 Statement:-Tangent at any point of a circle is perpendicular to the radius through the point of contact
If yes enter ( 1, ) else 0
10
59 Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal. 9
60 Perimeter of a circle is called as:
A . area
B. circumference
c. volume
D. none
9
61 The length of the common chord of two intersecting circles is ( 30 mathrm{cm} ). If the radii
of the two circles are ( 25 mathrm{cm} ) and ( 17 mathrm{cm} )
find the distance (in cm) between their
centres.
9
62 ( O ) is the center of the circle. ( O P=12 )
( mathrm{cm}, ) and ( boldsymbol{O B}=mathbf{1 3} mathrm{cm} . ) Find ( boldsymbol{A B} )
A. ( 8 mathrm{cm} )
B. ( 10 mathrm{cm} )
( mathrm{c} cdot 12 mathrm{cm} )
D. ( 13 mathrm{cm} )
9
63 Fill in the blanks:
The diameter of a circle are
9
64 In the given figure, ( O ) is the centre of
circle, ( angle A E C=40^{circ}, ) then find the value
of ( a+b+c )
9
65 In the diagram ( O ) is the centre of the
circle with diameter ( 20 mathrm{cm} )
The circle is the locus of a point ( boldsymbol{X} ) State
the distance of ( X ) from ( O )
( A cdot 5 mathrm{cm} )
B. ( 8 mathrm{cm} )
( c .10 mathrm{cm} )
D. ( 20 mathrm{cm} )
9
66 If ( 9.2 mathrm{cm} ) is the diameter of the circle,
then its radius is
A ( .4 .1 mathrm{cm} )
B. ( 4.6 mathrm{cm} )
c. ( 4.8 mathrm{cm} )
D. ( 4.3 mathrm{cm} )
9
67 The locus of the mid points of the chord
of the circle ( x^{2}+y^{2}=4, ) which
subtended a right angle at the origin is
A. ( x+y=1 )
B . ( x^{2}+y^{2}=1 )
c. ( x+y=2 )
D. ( x^{2}+y^{2}=2 )
9
68 Find the distance of a perpendicular from the centre of a circle to the chord if
the diameter of the circle is ( 30 mathrm{cm} ) and
its chord is ( 24 mathrm{cm} )
( mathbf{A} cdot 6 mathrm{cm} )
в. ( 7 mathrm{cm} )
( mathrm{c} .9 mathrm{cm} )
D. ( 10 mathrm{cm} )
9
69 The length of the chord ( x+y=3 )
intercepted by the circle ( x^{2}+y^{2}- )
( 2 x-2 y-2=0 ) is
A ( cdot frac{7}{2} )
B. ( frac{3 sqrt{3}}{2} )
c. ( sqrt{14} )
D. ( frac{sqrt{7}}{2} )
9
70 The radius of any circle touching the ( operatorname{lines} 3 x-4 y+5=0 ) and ( 6 x-8 y- )
( mathbf{9}=mathbf{0} ) is
A ( cdot frac{19}{10} )
в. ( frac{19}{20} )
c. ( frac{9}{20} )
D. ( frac{90}{20} )
10
71 Find the value of ( x )
( A, x=6 )
B . ( x=7 )
c. ( x=8 )
( x=9 )
9
72 If two parallel chords of length ( 8 mathrm{cm} ) and
( 6 mathrm{cm} ) in a circle of radius ( 5 mathrm{cm} ) are on
the opposite sides of the center then the
distance between the parallel chords is
A . ( 5 mathrm{cm} )
в. 6 ст
( c .7 mathrm{cm} )
D. None of these
9
73 If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel. 9
74 If tangents ( boldsymbol{P} boldsymbol{A} ) and ( boldsymbol{P} boldsymbol{B} ) from a point ( boldsymbol{P} )
to a circle with centre ( O ) are inclined to
each other at angle of ( 80^{circ}, ) then ( angle P O A ) is equal to.
A ( .50^{circ} )
B. ( 60^{circ} )
( c cdot 70^{circ} )
D. ( 80^{circ} )
10
75 Find the centre and radius of each of the
following circle. ( (x-5)^{2}+(y-3)^{2}=20 )
9
76 Three schools situated at ( P, Q ) and ( R ) in the figure are equidistant from each other as shown in the figure. Find
( angle Q O R )
9
77 Which one of the following statement is
true for the given circle?
A ( cdot overline{C D} cong overline{A B} )
B. ( overline{C D} neq overline{A B} )
c. ( widehat{C D} cong overline{A B} )
( mathbf{D} cdot overline{C D} cong overline{A B} )
9
78 A circle touches the hypotenuse of a right-angled triangle at its middle point and passes through the mid-point of the shorter side. If ( a ) and ( b(a<b) ) be the
length of the sides, then prove that the radius is ( frac{b}{4 a} sqrt{a^{2}+b^{2}} )
10
79 Prove that the centre of a circle
touching two intersecting lines lies on the angle bisector of the lines.
10
80 What is ( angle D A E ) from the figure, if
( B C=D E=5 ) and ( angle B A C=45^{circ} ? )
A . ( 30^{circ} )
B . ( 45^{circ} )
( c cdot 50^{circ} )
D. ( 60^{circ} )
9
81 Tangents TP and TO are drawn from a
point ( mathbf{T} ) to the circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=mathbf{a}^{2} . ) If the
point ( mathbf{T} ) lies on the line ( mathbf{p x}+mathbf{q y}=mathbf{r} )
then the locus of centre of the
circumcircle of ( Delta ) TPO is
A ( cdot p x+q y=frac{r}{3} )
B. ( p x+q y=frac{r}{2} )
c. ( p x+q y=2 r )
D. ( p x+q y=r )
10
82 Define the following term of the circle. Chord 9
83 In the given figure ‘O’ is the centre of the
circle and ( A B, C D ) are equal to chords. If
( <A O B=70^{circ} k . ) Find the angles of
( triangle O C D )
9
84 Two distinct chords drawn from the
point ( (p, q) ) on the circle ( x^{2}+y^{2}= )
( p x+q y ) are bisected at the ( x ) -axis.
Then
A ( cdot|p|=|q| )
B ( cdot p^{2}=8 q^{2} )
( mathbf{c} cdot p^{2}8 q^{2} )
9
85 Find the value of ( q )
( A cdot 3 sqrt{5} )
B. ( 3 sqrt{10} )
( c cdot 2 sqrt{10} )
D. ( 8 sqrt{10} )
10
86 The angle between tangents drawn from
the point (-1,3) to the circle ( x^{2}+ )
( boldsymbol{y}^{2}=mathbf{5} ) is
A.
в.
c.
D.
9
87 Given, a circle with designated center
designated perpendicular and radius 5
units. Find the length of the segment
labeled ( boldsymbol{x} )
( A cdot 4 )
B. 5
( c cdot 8 )
D. ( sqrt{10} )
E ( . sqrt{3} )
9
88 Find the value of ( J K ) in the following
figure if ( angle boldsymbol{H} boldsymbol{L} boldsymbol{G}=angle boldsymbol{J} boldsymbol{L} boldsymbol{K} )
A . 11
3
( c cdot 21 )
D.
9
89 What is chord? 9
90 A line that intersects a circle at two
distinct points is called
A . a diameter
B. a secant
c. a tangent
D. a radius
10
91 The circumference of the circle is
calculated by the formula
( mathbf{A} cdot 4 pi r )
В. ( 2 pi r^{2} )
c. ( 2 pi r )
D. ( pi r^{2} )
9
92 A tangent ( P Q ) at a point ( P ) of a circle of
radius ( 5 mathrm{cm} ) meets a line through the
centre ( O ) so that ( O Q=13 mathrm{cm} . ) Find the
length of ( P Q )
10
93 The internal centre of similitude of two
circles ( (x-3)^{2}+(y-2)^{2}= )
( mathbf{9},(boldsymbol{x}+mathbf{5})^{2}+(boldsymbol{y}+mathbf{6})^{2}=mathbf{9} ) is
A ( cdot(-1,-2) )
B. (-2,-1)
c. (3,2)
(年. (3,2)
D. (-5,-6)
9
94 The perpendicular from the centre of a circle to a chord bisects the chord. 9
95 Find the radius of the circle which
passes through the origin, (0,4) and (4,0)
A. ( sqrt{8} )
B. 4
c. 16
D. ( sqrt{36} )
9
96 Draw a pair of tangents to a circle of
radius 5 cm which are inclined to each
other at an angle of 60
10
97 In the figure, the radius of the smaller
circle is 3 centimetres, that of the
bigger circle is 6 centimetres and the distance between the centres of the
circles is 15 centimetres. PQ is a
tangent to both the circles. Find its
length.
10
98 Equal chords of a circle subtend equal
angle on centre
A . True
B. False
9
99 Define congruent chords. 9
100 If the length of the largest chord of a circle is ( 17 mathrm{cm}, ) find the radius of a circle.
( A cdot 34 mathrm{cm} )
B. ( 8.5 mathrm{cm} )
c. ( sqrt{17} mathrm{cm} )
D. ( sqrt{34} mathrm{cm} )
9
101 Draw the two tangents from a point
which is ( 9 mathrm{cm} ) away from the centre of a
circle of radius ( 3 mathrm{cm} ). Also, measure the
lengths of the tangents.
10
102 ( f angle C=angle D=50^{circ}, ) then four points ( A, B )
( C, D: )
A. Are con-cyclic
B. Do not lie on same circle
c. Are collinear
A,B.D and A,B,C lie on different circles
9
103 The line ( x=y ) touches a circle at the
point ( (1,1) . ) If the circle also passes through the point (1,-3) then its radius is:
begin{tabular}{l}
A ( .3 sqrt{2} ) \
hline
end{tabular}
B. 3
c. ( 2 sqrt{2} )
D. 2
10
104 If the tangents ( P A ) and ( P B ) are drawn from the point ( mathbf{P}(-1,2) ) to the circles ( x^{2}+y^{2}+x-2 y-3=0 ) and ( C ) is the
centre of the circle, then the area of the
quadrilateral PACB is
A .4
B. 16
c. does not exist
D. 12
10
105 In the diagram, ( A, B, C, D, E ) are points
on the circle. ( A B | D C, angle A D E=39^{circ} )
and ( angle A B C=62^{circ} . ) Then the values of
and ( y ) respectively are:
A ( cdot 23^{circ}, 51^{circ} )
B ( cdot 79^{circ}, 62 )
( c cdot 62^{circ}, 79^{circ} )
( 0.51^{circ}, 23 )
9
106 In the given figure, ( P A ) and ( P B ) are
tangents from an external point ( boldsymbol{P} ) to a
circle with center ( O . L N ) touches the
circle at ( M . ) Prove that ( boldsymbol{P} boldsymbol{L}+boldsymbol{L} boldsymbol{M}= )
( boldsymbol{P} boldsymbol{N}+boldsymbol{M} boldsymbol{N} )
10
107 Find the diameter of the circle if its.
Circumference is ( 62.8 mathrm{cm}(pi=3.14) )
9
108 n Fig.1, 0 is the centre of circle, ( A B ) is a
chord and ( A T ) is the tangent at ( A ). If
( angle A O B=100^{circ}, ) then ( angle B A T ) is equal to
A ( cdot 100^{circ} )
B. ( 40^{circ} )
( c cdot 50 )
D. 9 ?
10
109 The radius of the circle with centre at
the origin is 10 units. Write the
coordinates of the point where the circle intersects the axes. Find the distances
between any two of such points.
A ( . ) Co-ordinates ( =(10,0)(-10,0)(0,10)(0,-10) ) Distance ( =20,10 sqrt{2} ) units
B. ( C o- ) ordinates ( =(10,0)(-10,0)(0,10)(0,-10) ) Distance ( =10,10 sqrt{2} ) units
c. ( C o- ) ordinates ( =(10,0)(-10,0)(0,10)(0,-10), 0 ) Distance ( =20 sqrt{2} ) or ( 10 sqrt{2} ) units
D. none
9
110 The lines ( 3 x+4 y=9 ) and ( 6 x+8 y+ )
( mathbf{1 5}=mathbf{0} ) are tangents to the same circle.
The radius of the circle is :-
A ( cdot frac{3}{10} )
в. ( frac{33}{20} )
( c cdot frac{33}{10} )
D. ( frac{33}{5} )
10
111 In the adjoining figure ( A O B ) is a
diameter ( M P Q ) is a tangent at ( P ) then
the value of ( angle M P A ) is equal to
A ( cdot 25 )
B ( .26^{circ} )
( c cdot 27^{circ} )
( D cdot 30^{circ} )
10
112 A pair of opposite sides of a cyclic quadrilateral are equal. Prove that its diagonal are also equal 9
113 Two parallel chords are drawn in a circle
of diameter ( 30.0 mathrm{cm} . ) The length of one
chord is ( 24.0 mathrm{cm} ) and the distance
between the two chords is ( 21.0 mathrm{cm} . ) find
the length of the other chord.
9
114 Tangents ( P A ) and ( P B ) are drawn from
an external point ( P ) to two concentric
circle with centre ( O ) and radii ( 8 mathrm{cm} ) and
5 ( mathrm{cm} ) respectively, as shown in figure. If
( A P=15 mathrm{cm}, ) then find the length of
( boldsymbol{B P} )
10
115 CP and ( mathrm{CQ} ) are tangents to a circle with
centre ( 0 . A R B ) is another tangent
touching the circle at ( mathrm{R} . ) If ( C P= )
( 11 c m, B C=7 c m, ) then the length BR is
( A cdot 11 c m )
B. ( 7 mathrm{cm} )
( c .3 c m )
( mathrm{D} cdot 4 mathrm{cm} )
10
116 Through a fixed point ( (h, k) ) secants are
drawn to the circle ( x^{2}+y^{2}=r^{2} ) Then
the locus of the midpoints of the chords
intercepted by the circle is
A ( cdot x^{2}+y^{2}=h x+k y )
B . ( x^{2}-y^{2}=h x+k y )
C. ( x^{2}+y^{2}=h x-k y )
D. ( x^{2}-y^{2}=h x-k y )
10
117 In a circle with center ( O, ) a chord ( P Q ) is
such that ( boldsymbol{O} boldsymbol{M} pm boldsymbol{P} boldsymbol{Q} ) meeting ( boldsymbol{P} boldsymbol{Q} ) at ( boldsymbol{M} )
Then
( ^{mathbf{A}} cdot O Q^{2}=O M^{2}+frac{1}{2} P Q^{2} )
B. ( O Q^{2}=O M^{2}+frac{1}{4} P Q^{2} )
c. ( M Q^{2}=O M^{2}-O Q^{2} )
D. ( O M^{2}=M Q^{2}-O Q^{2} )
9
118 If the squares of the lengths of the tangents from a point ( P ) to the circles ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{a}^{2}, boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{b}^{2} ) and ( boldsymbol{x}^{2}+ )
( y^{2}=c^{2} operatorname{are} ) in A.P., then
This question has multiple correct options
A ( cdot a^{2}, b^{2}, c^{2} ) are in A.P
B. ( frac{1}{a^{2}}, frac{1}{b^{2}}, frac{1}{c^{2}} ) are in ( mathrm{H.P} )
c. ( a^{2}, b^{2}, c^{2} ) are in G.P
D. ( frac{1}{a^{2}}, frac{1}{b^{2}}, frac{1}{c^{2}} ) are in A.F.
10
119 statement-l: rrom a poınt ( boldsymbol{r} ) on tne
circle with centre ( boldsymbol{O} ) the chord ( boldsymbol{P} boldsymbol{A}=mathbf{8} )
( mathrm{cm} ) is drawn. The radius of the circle is
( 24 mathrm{cm} . ) Let ( P B ) be drawn parallel to ( O A )
Suppose ( B O ) extended meet ( P A )
extended at ( M . ) The length of ( M A ) is 9
( mathrm{cm} )
Reason
Statement-2: ( O A ) is a radius of a circle
with centre at ( O . R ) is a point on ( O A )
through which a chord ( C D )
perpendicular to ( boldsymbol{O} boldsymbol{A} ) is drawn. Let a chord through A meet the chord ( C D ) at
( M ) and the circle at ( B ). Also ( O S ) is
perpendicular from ( boldsymbol{O} ) on chord ( boldsymbol{A} boldsymbol{B} ). The radius of the circle is ( 18 mathrm{cm} . R ) is the mid point of ( boldsymbol{A O} ) and ( boldsymbol{A} boldsymbol{M} / boldsymbol{M} boldsymbol{B}=frac{mathbf{1}}{mathbf{2}} )
The length of ( boldsymbol{O} boldsymbol{S} ) is ( boldsymbol{9} mathrm{cm} )
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
9
120 The radius of a circle is ( 13 mathrm{cm} ) and the
length of one of its chords is ( 10 mathrm{cm} ). The distance of the chord from the centre is
A. ( 8 mathrm{cm} )
B. ( 10 mathrm{cm} )
( mathrm{c} cdot 12 mathrm{cm} )
D. ( 6 mathrm{cm} )
9
121 In Fig. 3 , if 0 is the centre of the circle
( mathrm{OL}=4 mathrm{cm}, mathrm{AB}=6 mathrm{cm} ) and ( mathrm{OM}=3 )
( mathrm{cm}, ) then ( mathrm{CD}=? )
( A cdot 4 c m )
в. ( 8 mathrm{cm} )
( c . ) 6cm
D. 10cm
9
122 n fig, chord ( A B | C D ) of a circle
with centre ( O ) and radius 5 cm such
that ( A B=6 mathrm{cm} ) and ( C D=8 mathrm{cm} . ) if ( O P )
( perp A B, O Q . perp C D, ) then ( P Q ) in cm is
( mathbf{A} cdot 4 mathrm{cm} )
( mathbf{B} cdot 7 mathrm{cm} )
( c cdot 1 mathrm{cm} )
D. ( 3 mathrm{cm} )
9
123 Find the centre and radius of the circle
( x^{2}+y^{2}-4 x-8 y-45=0 )
9
124 In the given diagram, ( O ) is the centre of
the circle and ( P Q R ) is a straight line.
The value of ( x ) is
A ( cdot 110^{circ} )
B. ( 120^{circ} )
( c cdot 130^{circ} )
D. ( 140^{circ} )
9
125 Recall that two circles are congruent if
they have the same radius then equal chords of congruent circles subtend equal angles at their centres.
A . True
B. False
9
126 Each of the height and radius of the base of a right circular cone is
increased by ( 100 % ). The volume of the cone will be increased by
A . ( 700 % )
в. ( 500 % )
c. ( 300 % )
D. ( 100 % )
9
127 ( A B C D ) is a cyclic quadrilateral such
that ( A B ) is a diameter of the circle
circumscribing it and ( angle A D C=140^{circ} )
then ( angle B A C ) is equal to
( A cdot 80 )
B. ( 50^{circ} )
( c cdot 40^{circ} )
D. ( 30^{circ} )
9
128 A circular area having a radius ( 20 mathrm{cm} ) is divided into two equal parts by a concentric circle of radius ‘r’. The value
of ‘r’ will be
A. ( 5 mathrm{cm} )
B. 10 ( mathrm{cm} )
( mathrm{c} cdot 5 sqrt{2} mathrm{cm} )
D. ( 10 sqrt{2} mathrm{cm} )
9
129 69. The circumcentre of a triangle
ABC is O. If Z BAC = 85° and
BCA = 75°, then the value of
2 OAC is
(1) 40° (2) 60°
(3) 70° (4) 90°
9
130 Line segment joining the centre to any point on the circle is
A. radius of the circle
B. diameter of the circle
c. secant of the circle
D. tangent of the circle.
9
131 In the given figure points ( A, D, P, C ) and ( B ) lie on a circle with centre
( boldsymbol{O}, angle boldsymbol{B O D}=mathbf{1 5 0}^{circ} ) Find the measures
of ( angle B P D, angle B C D ) and ( angle B A D )
9
132 Equation of a straight line meeting the circle ( x^{2}+y^{2}=100 ) in two points each
point at a distance of 4 from the point (8,6) on the circle is
A. ( 4 x+3 y-50=0 )
B. ( 4 x+3 y-100=0 )
c. ( 4 x+3 y-46=0 )
D. none of these
9
133 If the line ( h x+k y=1 ) touches ( x^{2}+ )
( y^{2}=a^{2}, ) then the locus of the point ( (h )
k) is a circle of radius
( A cdot a )
B.
( c cdot sqrt{a} )
D. ( frac{1}{sqrt{a}} )
9
134 what is tangent of a circle and
definition?
10
135 If the common chord of the circle ( x^{2}+ )
( (y-lambda)^{2}=16 ) and ( x^{2}+y^{2}=16 )
subtend a right angle at the origin then ( lambda ) is equal to
( A cdot 4 )
B. ( 4 sqrt{2} )
( c cdot pm 4 sqrt{2} )
( D .8 )
9
136 A and ( mathrm{B} ) are two points on the circle
( mathbf{x}^{2}+mathbf{y}^{2}=1 . ) If the ( mathbf{x} ) co-ordinates of ( mathbf{A} )
and ( mathrm{B} ) are the roots of the equation
( x^{2}+a x+b=0 ) and the ( y )
coordinates of ( mathbf{A} ) and ( mathbf{B} ) are the roots of
the equation ( mathbf{y}^{2}+mathbf{b y}+mathbf{a}=mathbf{0} ) then the
equation of the line ( A B ) is
A ( cdot a x+b y=0 )
B. ( a x+b y+1=0 )
c. ( b x+a y+a+b=0 )
D. ( a x+b y+a+b+1=0 )
9
137 The tangents drawn at the ends of a diameter of a circle are ?
A. perpendicular
B. parallel
c. adjacent
D. none of the above
10
138 In the given figure, ( Delta X Y Z ) is inscribed
in a circle with centre 0. If the length of chord YZ is equal to the radius of the
circle OY then ( angle boldsymbol{Y} boldsymbol{X} boldsymbol{Z}= )
A ( cdot 60^{circ} )
B. ( 30^{circ} )
( c cdot 80^{circ} )
D. ( 100^{circ} )
9
139 In figure, ( boldsymbol{K} boldsymbol{X} boldsymbol{M} ) is a tangent to the
circumcircle ( C ) of ( triangle X Y Z ) such that
( boldsymbol{L} boldsymbol{M} | boldsymbol{Y} boldsymbol{Z} . ) Show that ( boldsymbol{X} boldsymbol{Y}=boldsymbol{X} boldsymbol{Z} )
9
140 In the figure, 0 is the centre of the circle
Find the length of ( mathrm{CD} ), if ( mathrm{AB}=5 mathrm{cm} )
9
141 69. O is the circumcentre of A ABC.
If Z BAC = 85°, Z BCA = 75°,
then 2 OAC is equal to
(1) 70°
(2) 60°
(3) 80° (4) 100°
9
142 Assertion
The circle of smallest radius passing through two given points ( A & B ) must be of radius ( frac{1}{2} A B )
Reason
A straight line is a shortest distance between two points.
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
9
143 The square of the length of the tangent
from (3,-4) to the circle ( x^{2}+y^{2}- )
( 4 x-6 y+3=0 ) is
A . 20
B. 30
( c cdot 40 )
D. 50
10
144 For each ( k epsilon N, ) let ( C_{k} ) denote the circle
whose equation is ( x^{2}+y^{2}=k^{2} . ) On the
circle ( C_{k}, ) a particle moves k units in the
anticlockwise direction. After
completing its motion on ( C_{k}, ) the
particle moves to ( C_{k+1} ) in the radial direction. The motion of the particle continues in this manner. The particle starts at ( (1,0) . ) If the particle crossed the positive direction of the ( x ) -axis for
the first time of the circle ( C_{n} ) then ( n ) is
A . 7
B. 6
( c cdot 2 )
D. none of these
9
145 In the figure, if ( boldsymbol{O A}=mathbf{1 7 c m}, boldsymbol{A B}= )
( 30 mathrm{cm} ) and ( 0 mathrm{D} ) is perpendicular to ( mathrm{AB} )
then ( mathrm{CD} ) is equal to:
( A cdot 8 c m )
в. ( 9 mathrm{cm} )
( c cdot 10 c m )
D. ( 11 mathrm{cm} )
9
146 If ( bar{P} Q ) is a chord of a circle with centre ( O )
and ( P R ) is a tangent to the circle at ( P )
then ( angle P O Q= )
A. ( 4 angle R P Q )
B. ( 3 angle R P Q )
c. ( 2 angle R P Q )
D. ( angle R P Q )
9
147 State the following statement is True or
False

If the chords of a circle intersect within
a circle, then the rectangle formed by the parts of the same chord have equal
area
A. True
B. False

9
148 From the following figure, choose the
statements that are correct.
i) Congruent chords have congruent
( operatorname{arcs} )
ii) Congruent chords have equal centra
angles.
iii) Congruent arcs have congruent
central angles.
iv) Chords equidistant from the center
are congruent.
A. ii and iii only
B. iii and iv
D. All of the above
9
149 Show that all the chords of the curve
( 3 x^{2}-y^{2}-2 x+4 y=0 ) which subtend
a right angle at the origin?
9
150 The diameter of the circle is ( 2 mathrm{cm} ). What
is the circumference?
( mathbf{A} cdot 12.28 mathrm{cm} )
B. ( 6.2 mathrm{cm} )
c. ( 18.28 mathrm{cm} )
D. ( 10.28 mathrm{cm} )
9
151 In a circle if a chord ( A B ) is nearer to the
centre ( boldsymbol{O} ) than the chord ( boldsymbol{C} boldsymbol{D} ) then:
( mathbf{A} cdot A B>C D )
B. ( A B=C D )
c. ( A B<C D )
D. none of these
9
152 Define diameter. 9
153 Prove that the centre of the smallest
circle passing through origin and whose centre lies on ( boldsymbol{y}=boldsymbol{x}+mathbf{1} ) is ( left(-frac{mathbf{1}}{mathbf{2}}, frac{mathbf{1}}{mathbf{2}}right) )
9
154 68. The length of a chord of a circle
is equal to the radius of the cir-
cle. The angle which this chord
subtends in the major segment
of the circle is equal to
(1) 30°
(2) 45°
(3) 60°
(4) 90°
9
155 In the figure, ( boldsymbol{O} ) is the point of
intersection of two chords ( A B ) and ( C D )
such that ( O B=O D ), then triangles
( O A C ) and ( O D B ) are:
A. Equilateral but not similar
B. Isosceles but not similar
c. Equilateral and similar
D. Isosceles and similar
9
156 A circle touches the sides of a
quadrilatieral ABCD at P, Q, R, S respectively. The angles subtended at the centre by a pair of opposite sides have theirs sum as:
10
157 Chords of the circle ( x^{2}+y^{2}+2 g x+ )
( 2 f y+c=0 ) subtends a right angle at
the origin. The locus of the feet of the perpendiculars from the origin to these chords is
A ( cdot x^{2}+y^{2}+g x+f y+c=0 )
B . ( 2left(mathrm{x}^{2}+mathrm{y}^{2}right)+mathrm{gx}+mathrm{fy}+mathrm{c}=0 )
C ( cdot 2left(mathrm{x}^{2}+mathrm{y}^{2}+mathrm{gx}+mathrm{fy}right)+mathrm{c}=0 )
D. ( x^{2}+y^{2}+2(g x+f y+c)=0 )
9
158 Find the centres of the circles passing through (-4,3) and touching the lines ( boldsymbol{x}+boldsymbol{y}=boldsymbol{2} ) and ( boldsymbol{x}-boldsymbol{y}=boldsymbol{2} )
A ( cdot((-10 pm sqrt{54}), 0) )
B. ( (10 pm sqrt{54}, 0) )
c. ( (0,-10 pm sqrt{54}) )
D. ( (0,10 pm sqrt{54}) )
10
159 Find the length of a chord which is at a distance of ( 3 mathrm{cm} ) from the centre of a
circle of radius ( 5 mathrm{cm} )
A . ( 2 mathrm{cm} )
B. ( 6 mathrm{cm} )
( c cdot 8 mathrm{cm} )
D. ( 10 mathrm{cm} )
9
160 If ( angle R P Q=45^{circ}, ) then find ( angle P Q R )
( mathbf{A} cdot 15^{circ} )
B ( .30^{circ} )
( c cdot 60^{circ} )
D ( .45^{circ} )
10
161 Prove that if chords of congruent circles subtend equal angle at their centres, then the chords are equal. 9
162 Find the value of ( x+y ) in the given
figure (in degrees)
10
163 From the figure, identify a sector 9
164 Recall that two circles are congruent if they have the same radii. Prove that
equal chords of congruent circles subtend equal angles at their centres.
9
165 In a circle of diameter ( 10 mathrm{cm} ), the length of each of 2 equal and parallel chords is ( 8 mathrm{cm}, ) then the distance between these
two chords is
A. ( 4 mathrm{cm} )
B. ( 5 mathrm{cm} )
( c cdot 6 mathrm{cm} )
D. ( 7 mathrm{cm} )
9
166 Draw any circle and mark an arc. 9
167 Find the distance of a perpendicular
from the centre of a circle to the chord if
the diameter of the circle is ( 30 mathrm{cm} ) and
its chord is ( 24 mathrm{cm} )
A ( .6 mathrm{cm} )
B. ( 7 mathrm{cm} )
( c .9 mathrm{cm} )
D. ( 10 mathrm{cm} )
9
168 If the chord ( y=m x+1 ) of the circle
( x^{2}+y^{2}=1 ) subtends an angle of
measure ( 45^{circ} ) at the major segment of
the circle then the value of ( m ) is
A ( .2 pm sqrt{2} )
B. ( -2 pm sqrt{2} )
c. ( -1 pm sqrt{2} )
D. none of these
9
169 In given figure, ( P Q ) is chord of length
( 8 c m ) of a circle of radius ( 5 c m, ) the
tangents at ( P ) and ( Q ) intersect at a point
T. Find the length ( boldsymbol{T} boldsymbol{P} )
9
170 A secant intersects the circle at
points.
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
( D )
10
171 Tangents are drawn to the circle ( x^{2}+ )
( y^{2}=25 ) from the point ( (13,0) . ) Prove
that the angle between them is ( 2 tan ^{-1}(5 / 12) ) and their equations are
( 12 y+5 x+65=0 ) and ( 12 y-5 x- )
( mathbf{6 5}=mathbf{0} )
10
172 66. Two chords AB and CD of a cir-
cle with centre O intersect
each other at the point P. If
ZAOD = 20° and ZBOC = 30°,
then ZBPC is equal to:
(1) 50°
(2) 20°
(3) 25°
(4) 30°
9
173 The length of a chord of a circle of
radius ( 10 mathrm{cm} ) is ( 12 mathrm{cm} ). Find the
distance of the chord from the centre of
the circle
A. ( 6 mathrm{cm} )
B. ( 5 mathrm{cm} )
( c .8 mathrm{cm} )
D. ( 7 mathrm{cm} )
9
174 Find the equation of the tangent to the
curve ( y=x^{2}-7 ) at the point where it
cuts the ( y ) – axis.
10
175 Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at a
center.
10
176 The distance between two parallel chords, each of length 10 units is 24 units then the radius of the circle is:
A . 5
B. 12
c. 13
D. 30
9
177 ( O ) is the centre of the circle. ( A B ) and
( C D ) are two chords of the circle. ( O M perp )
( A B ) and ( O N perp C D . ) If ( O M=O N=3 )
( operatorname{cmand} A M=B M=4.5 mathrm{cm}, ) then ( C D )
is equal to
( A cdot 8 mathrm{cm} )
B. 9 cm
( c .10 mathrm{cm} )
D. None of these
9
178 In the given figure, 0 is the centre of the
circle and XOY is a diameter. If ( X Z ) is
any other chord of the circle, then which
of the following is correct?
( A cdot X Zx z )
( c cdot 0 x+0 z )
D. zX + ZY
9
179 A boat in a circular lake lies at its
centre. The perpendicular distance of the boat is ( 10 mathrm{m} ) from a bridge lying in
( 40 mathrm{m} ) distance across the circular lake.
Find the distance that the boat will have
to travel to reach to the extreme point of left side of bridge is ( m sqrt{5} mathrm{m} ). Then, ( m ) is
A . ( 20 mathrm{m} )
в. ( 10 mathrm{m} )
( c .-10 m )
D. both B &
9
180 Find the center and radius of the circle.
( (x+5)^{2}+(y-3)^{2}=36 )
9
181 What is the angle between the line joining the centre and point of contact of a tangent and the tangent itself?
( mathbf{A} cdot mathbf{0} )
B . 45
( c .90 )
D. ( 180^{circ} )
10
182 The radius of a circle is ( 40 mathrm{cm} ) and the length of perpendicular drawn from its centre to chord is ( 24 mathrm{cm} . ) The length of
the chord ( A B ) is
A. ( 32 mathrm{cm} )
B. 64cm
c. ( 48 mathrm{cm} )
D. 24cm
9
183 ( l x+m y+n=0 ) is a tangent line to
the circle ( x^{2}+y^{2}=r^{2}, ) if
A ( cdot l^{2}+m^{2}=n^{2} r^{2} )
B . ( l^{2}+m^{2}=n^{2}+r^{2} )
C ( cdot n^{2}=r^{2}left(l^{2}+m^{2}right) )
D. none of these
10
184 circle is a
69. Two cu
Two circles of diameters 10
and 6 cm have the same cene
A chord of the larger circle
tangent of the smaller one,
length of the chord is
(1) 4 cm. (2) 8 cm.
(3) 6 cm. (4) 10 cm
9
185 62. The diagonals AC and BD of a
cyclic quadrilateral ABCD inter-
sect each other at the point P.
Then, it is always true that
(1) BP. AB = CD. CP
(2) AP. CP = BP. DP
(3) AP.BP = CP. DP
(4) AP. CD = AB .CP
9
186 A point ( P ) is outside a circle at a
distance of ( 13 mathrm{cm} ) from its centre.
secant from ( P ) cuts the circle in ( Q ) and
( R ) such that ( Q R=7 mathrm{cm} ) and the
segment ( P Q ) of the secant, exterior to
the circle is ( 9 mathrm{cm} . ) Therefore, the radius
of circle is
A. 3 cm
в. 4 ст
( c .5 mathrm{cm} )
D. ( 6 mathrm{cm} )
9
187 In the given figure, ( A B ) is a chord of
length ( 16 mathrm{cm} ) of a radius ( 10 mathrm{cm} . ) The
tangents at ( A ) and ( B ) intersect at point
( P . ) Find the length of ( P A )
10
188 The equation of the circle and its chord are respectively ( x^{2}+y^{2}=a^{2} ) are
( x cos alpha+y sin alpha=p . ) The equation of
the circle of which this chord
is diameter is
A ( cdot x^{2}+y^{2}-2 p x cos alpha-2 p y sin alpha+2 p^{2}-a^{2}=0 )
B . ( x^{2}+y^{2}-2 p x cos alpha-2 p y sin alpha+p^{2}-a^{2}=0 )
C ( cdot x^{2}+y^{2}-2 p x cos alpha+2 p y sin alpha+2 p^{2}-a^{2}=0 )
D. None of these
9
189 A point ( P ) is ( 13 mathrm{cm} ) from the centre of
the circle. The length of the tangent
drawn from ( P ) to the circle is 12 cm.
Find the radius of the circle.
10
190 Write True or False and justify your answer in each of the following:
If a number of circles touch a given line segment PQ at a point ( A, ) then their centres lie on the perpendicular bisector of PQ.
A. True
B. False
c. Ambiguous
D. Data Insufficient
9
191 Illustration 2.21 Find the length of an arc of a circle of
radius 5 cm subtending a central angle measuring 15º.
9
192 n Fig. 0 is the centre of the circle such
that ( angle A O C=130^{circ}, ) then ( angle A B C= )
A ( cdot 130 )
B. 115
( c cdot 65 )
D. 165
9
193 Two parallel chords in a circle are ( 10 mathrm{cm} 10 mathrm{cm} ) and ( 24 mathrm{cm} 24 mathrm{cm} ) long. If
the radius of the circle is ( 13 mathrm{cm} 13 mathrm{cm} ) find the distance between the chords if
thay lie on the same side of the centre.
9
194 n Fig. ( A B ) and ( C D ) are common tangents
to two circles of unequal radii then ( A B )
is not equal to ( mathrm{CD} )
A. True
3. Falss
10
195 STATEMENT – 1: The locus of the middle
points of equal chords of a circle with
centre at 0 is a circle with centre at 0
STATEMENT – 2 : The mid point of the equal chords are equidistant from the centre of the circle.
A. Statement – 1 is True, Statement – 2 is True, Statement 2 is a correct explanation for Statement – 1
B. Statement – 1 is True, Statement – 2 is True : Statement 2 is NOT a correct explanation for Statement- –
c. Statement – 1 is True, Statement – 2 is False
D. Statement-1 is False, Statement- – 2 is True
9
196 A tangent is drawn to each of the
circles
( boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{a}^{2}, boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{b}^{2} )
Show that if the two tangents are
mutually perpendicular, the locus of their point of intersection is a circle concentric with the given circles.
10
197 ( A B ) and ( C D ) are two parallel chords of a circle of radius ( 3 mathrm{cm} . ) If ( A B=4 mathrm{cms} )
and ( C D=5 mathrm{cm} . ) Then the distance
between them (in ( mathrm{cm} ) ) is
A ( cdot frac{sqrt{5}}{2}+sqrt{11} )
B. ( sqrt{5}+sqrt{11} )
( ^{c} cdot sqrt{5}+frac{sqrt{11}}{2} )
D. ( sqrt{2}+frac{sqrt{11}}{sqrt{5}} )
9
198 Find the length of a chord which is at a
distance of ( 4 mathrm{cm} ) from the centre of a
circle whose radius is ( 5 mathrm{cm} )
9
199 The condition that the chord ( x cos alpha+ )
( boldsymbol{y} sin boldsymbol{alpha}-boldsymbol{p}=mathbf{0} ) of ( boldsymbol{x}^{2}+boldsymbol{y}^{2}-boldsymbol{a}^{2}=mathbf{0} ) may
subtend a right angle at the centre of the circle is
A ( cdot a^{2}=2 p^{2} )
B ( cdot p^{2}=2 a^{2} )
c. ( a=2 p )
D. ( p=2 a )
9
200 Three circles with centre ( A, B ) and ( C )
respectively, touch one another as
shown in the figure. If ( A, B ) and ( C ) are
collinear and PQ is a common tangent
to the two smaller circles, where ( mathrm{PQ}=4 )
the area of shaded region is
10
201 Find the centre and radius of the circle
( 2 x^{2}+2 y^{2}=3 x-5 y+7 )
9
202 Out of the two concentric circles, the
radius of the outer circle is ( 5 mathrm{cm} ) and
the chord ( A C ) of length ( 8 mathrm{cm} ) is a tangent to the inner circle. Find the radius of
the inner circle.
( A .3 c m )
B. ( 6 mathrm{cm} )
( c .5 mathrm{cm} )
D. 7 cm
9
203 The center of a circle which passes
through the points (0,0),(1,0) and touches the circles ( x^{2}+y^{2}=9 )
( ^{A} cdotleft(frac{3}{2}, frac{1}{2}right) )
в. ( left(frac{1}{2}, frac{3}{2}right) )
c. ( left(frac{1}{2}, frac{1}{2}right) )
D. ( left(frac{1}{2}, sqrt{2}right) )
9
204 What are the coordinates of the center
of the circle represented by the
equation ( (x+3)^{2}+(y-4)^{2}=25 ? )
( A cdot(3,4) )
B. (3,-4)
c. (-3,4)
D. (-3,-4)
9
205 If the area and the circumference of
circle are numerically equal, then the radius the circle is
( ^{A} cdot frac{5}{2} )
B. 2
c. 1
D. 2 ( overline{5} )
9
206 56. Each of the circles of equal radil
with centres A and B pass
through the centre of one anoth-
er circle they cut at C and D then
DBC is equal to
(1) 60° (2) 100
(3) 120°
(4) 140°
9
207 The common chord of ( x^{2}+y^{2}-4 x- )
( 4 y=0 ) and ( x^{2}+y^{2}=16 ) subtends at
the origin an angle to
A. ( pi / 6 )
в. ( pi / 4 )
c. ( pi / 3 )
D. ( pi / 5 )
9
208 ( operatorname{Let} O P=5 ) and ( P M=4 ) Find ( O M )
( A cdot 3 c m )
B. ( 4 mathrm{cm} )
( c .5 mathrm{cm} )
D. ( 8 c m )
9
209 If radii of two concentric circles are 4
( mathrm{cm} ) and ( 5 mathrm{cm}, ) then the length of each chord of one circle which is tangent to
the other circle is
A. ( 3 mathrm{cm} )
B. 6 ( mathrm{cm} )
( c cdot 9 mathrm{cm} )
D. ( 1 mathrm{cm} )
10
210 69. The length of a tangent from
an external point to a circle is
5/3 unit. If radius of the circle
is 5 units, then the distance of
the point from the circle is
(1) 5 units (2) 15 units
(3) -5 units (4) -15 units
10
211 Prove that the tangents drawn from an
external point to a circle are equal.
10
212 In the figure, ( M N S ) is tangent to the
circle with centre ( O ) at ( N )
( A B ) is chord parallel to ( M N S . ) find
( angle A N C )
A . 50
B. ( 90^{circ} )
( c cdot 40 )
D. 20
9
213 The equation of the circle, passing through the point ( (2,8), ) touching the ( operatorname{lines} 4 x-3 y-24=0 ) and ( 4 x-3 y- )
( 42=0 ) and having ( x ) coordinate of the
center of the circle numerically less
then or equal to 8 , is
A ( cdot x^{2}+y^{2}+4 x-6 y-12=0 )
B. ( x^{2}+y^{2}-4 x+6 y-12=0 )
C ( cdot x^{2}+y^{2}-4 x-6 y-12=0 )
D. None of these
10
214 The longest chord passes through a centre of a circle is 9
215 In the diagram, ( P ) is the centre of the
circle with radius ( 4 mathrm{cm} ) and ( Q ) is the
centre of the circle with radius ( 3 mathrm{cm} )
Of the points marked ( W, X, Y ) and ( Z )
which point is ( 4 mathrm{cm} ) from ( P ) and more
than ( 3 mathrm{cm} ) from ( Q ? )
( A cdot W )
в. ( x )
( c . Y )
D.
9
216 75. In the given figure, POg is a di-
ameter and PQRS is a cyclic
quadrilateral. If ZPSR = 130°,
then the value of ZRPO is
130°
(1) 30°
(3) 45°
(2) 40°
(4) 35°
9
217 In a circle with centre ( 0 . operatorname{seq} mathrm{PQ}, ) is a
chord such that ( angle P O Q=70^{circ} . ) Find the
( angle O P Q )
9
218 Suppose you are given a circle. Give steps of construction to find its centre. 9
219 n Fig. ( 2, ) ‘O’ is the centre of the circle,
find ( angle A O C, operatorname{given} angle B A O=30^{circ} ) and
( angle B C O=40^{circ} )
( A cdot 35 )
В. 140
( c cdot 70 )
D. Cannot be determined
9
220 In the given figure, a circle with centre ( O ) is given in which a diameter ( A B )
bisects the chord ( C D ) at a point ( E ) such
that ( C E=E D=8 c m ) and ( E B=4 c m )
Find the radius of the circle.
9
221 ACB is a tangent to a circle at c.
CD and CE are chords such that
ZACE > ZACD. If ZACD = ZBCE
= 50°. then :
(1) CD = CE
(2) ED is not parallel to AB
(3) ED passes through the cen-
tre of the circle
(4) A CDE is a right angled trian-
gle
9
222 Find the value of ( c ) if (2,3) lies on the circle ( x^{2}+y^{2}+2 x+3 y+c=0 ) 9
223 Prove: If a chord of circle ( x^{2}+y^{2}=8 )
makes equal intercepts of length ‘a’ on the coordinate axes then ( |a|<4 )
9
224 If ( O ) is the centre of a circle, ( P Q ) is a
chord and the tangent ( P R ) at ( P ) makes
an angle of ( 50^{0} ) with ( P Q ), then find the
Angle ( (P O Q) )
10
225 The tangents are drawn from origin and the point ( (boldsymbol{g}, boldsymbol{f}) ) to the circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+ )
( 2 g x+2 f y+c=0 . ) Find the distance
between chords of contact.
A ( cdot frac{2left(g^{2}+f^{2}-cright)}{sqrt{g^{2}+f^{2}}} )
B. ( frac{g^{2}+f^{2}-c}{sqrt{g^{2}+f^{2}}} )
c. ( frac{g^{2}+f^{2}-c}{2 sqrt{g^{2}+f^{2}}} )
D. none of these
9
226 The length of a tangent from a point ( boldsymbol{A} )
at distance ( 5 mathrm{cm} ) from the centre of the
circle is ( 4 mathrm{cm} ).Find the radius of the
circle.
9
227 O’ is the centre of the circle ( angle Q P S= )
( mathbf{6 5}^{circ} ; angle boldsymbol{P} boldsymbol{R} boldsymbol{S}=mathbf{3 3}^{circ}, )then ( angle boldsymbol{P S Q}= )
A .90
B. ( 82^{circ} )
( c cdot 102 )
( D cdot 42 )
9
228 Find the equations to the circles in which the line joining the points ( (a, b) ) and ( (b,-a) ) is a chord subtending an
angle of ( 45^{circ} ) at any point on its circumference.
9
229 Prove that the parallelogram
circumscribing a circle is a rhombus.
10
230 If the points (2,0),(0,1),(4,5) and ( (0, c) ) are concyclic, then the value of ( c ) is This question has multiple correct options
A . -1
B.
c. ( frac{14}{3} )
D. ( -frac{14}{3} )
9
231 In the given figure below, ( A D ) is a
diameter. ( O ) is the centre of the circle.
( A D ) is parallel to ( B C ) and ( angle C B D=32^{circ} )
Find ( angle B E Dleft(text { in }^{circ}right) )
9
232 73. PO is a tangent to the circle at R
then mZPRS is equal to :
BOT
(1) 30°
(3) 60°
(2) 40°
(4) 80°
9
233 The equation of the circle with center (1,2) and tangent ( x+y-5=0 ) is
A ( cdot x^{2}+y^{2}+2 x-4 y+6=0 )
B . ( x^{2}+y^{2}-2 x-4 y+3=0 )
c. ( x^{2}+y^{2}-2 x-4 y-8=0 )
D. ( x^{2}+y^{2}-2 x-4 y+8=0 )
10
234 In the diagram, PQ and QR are tangents
to the circle, centre ( 0, ) at ( P ) and ( R )
respectively. Find the value
( A cdot 25 )
3.35
( c cdot 45 )
55
10
235 Chord ( A B ) of the circle ( x^{2}+y^{2}=100 )
passes through the point (7,1) and
subtends an angle of ( 60^{circ} ) at the
circumference of the circle. If ( m_{1} ) and
( m_{2} ) are the slopes of two such chords
then the value of ( m_{1} m_{2} ) is
A . -1
B.
c. ( 7 / 12 )
D. –
9
236 A chord of a circle of radius ( 12 mathrm{cm} )
subtends an angle of ( 120^{0} ) at the centre.
Find the area of the corresponding segment
9
237 Find the radius of the circle passing the points (0,0),(1,0) and (0,1) 9
238 Tangent ( 0 A ) and ( O B ) are drawn for
( O(0,0) ) to the circle ( (x-1)^{2}+(y- )
1) ( ^{2}=1 )
Equation of the circumcircle of triangle
OAB 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. ( x^{2}+y^{2}-x-y=0 )
10
239 The coordinates of the fixed point of the
chord cut off by ( 2 x-5 y+18=0 ) by
the circle ( x^{2}+y^{2}-6 x+2 y-54=0 )
are
A ( .(1,4) )
в. (2,4)
c. (4,1)
()
D. (1,1)
9
240 59. Two circle with centres O and O’
touch externally each other at
point P. A straight line is drawn
from P which intersects both the
circles at Q and R. Given that
radii of the circles OP= 6 cm and
O’P=4 cm and PQ = 10 cm, then
PR = ?
(1) 7.6 cm (2) 7.8 cm
(3) 6.7 cm (4) 7.5 cm
10
241 If ( P ) is a point, then how many tangents to a circle can be drawn from the point ( P, ) if it lies On the circle.
( A cdot 0 )
B.
( c cdot 2 )
D. 3
10
242 71. The tangents at two points A
and B on the circle with cen-
tre O intersect at P: if in
quadrilateral PAOB, ZAOB:
ZAPB = 5:1, then measure
of ZAPB is :
(1) 30º (2) 60°
(3) 45° (4) 15°
9
243 Circumference of a circle is equal to
A . ( pi r )
в. ( 2 pi r )
c. ( frac{pi r}{2} )
D. ( 2+frac{pi r}{2} r )
9
244 f 0 is a point on the circle and ( P ) is a
point in the exterior of the circle. Length
of ( boldsymbol{O} boldsymbol{P}=7.5 mathrm{cm} ) and radius of the circle
is ( 5.5 mathrm{cm} . ) What will be the length of ( Q P ) if ( Q ) is the centre?
A. ( 5.5 mathrm{cm} )
B. ( 13 mathrm{cm} )
( c .7 .5 mathrm{cm} )
D. ( 13.5 mathrm{cm} )
9
245 Find diameter ( & ) circumference with
radius ( 7.7 mathrm{cm} )
9
246 In the figure ( A O C ) is a diameter of the circle and are ( overline{boldsymbol{A} times boldsymbol{B}}=frac{1}{2} overline{boldsymbol{B} boldsymbol{Y} boldsymbol{C}} ). Find
( angle B O C )
9
247 In the given figure, ( A O B ) is a diameter
of the circle with center ( boldsymbol{O} ) and ( boldsymbol{A} boldsymbol{C} ) is a
tangent to the circle at ( A ). If ( angle B O C= )
( 130^{circ}, ) then find ( angle A C O )
9
248 A circular park of radius ( 20 m ) is
situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.
A ( .5 sqrt{3} mathrm{cm} )
B. ( 2 sqrt{3} mathrm{cm} )
c. ( sqrt{3} mathrm{cm} )
D. ( 20 sqrt{3} mathrm{cm} )
9
249 The center of a circle represented by the
equation ( (x-2)^{2}+(y+3)^{2}=100 ) is
located in Quadrant
( A )
B. I
( c )
( D cdot|v| )
9
250 ( f(Omega A=5 mathrm{cm}, A B=8 mathrm{cm} text { and } O D ) is
perpendicular to ( A B, ) then ( C D ) is equal
to:
( A cdot 2 mathrm{cm} )
B. ( 3 mathrm{cm} )
( c .4 mathrm{cm} )
( 0.5 mathrm{cm} )
9
251 Find the total cost of wooden fencing around a circular garden of diameter ( 28 m . ) If ( 1 m ) of fencing costs 2300 9
252 The radius of a circle with centre ( boldsymbol{P} ) is
( 25 mathrm{cm} ) and the length of the chord is 48 ( mathrm{cm} . ) The distance of the chord from
centre ( P ) of the circle is
( mathbf{A} cdot 24 mathrm{cm} )
B. ( 5 mathrm{cm} )
( c .7 mathrm{cm} )
D. ( 12 mathrm{cm} )
9
253 Let ( A B C ) be an equilateral triangle
inscribed in circle ( 0 . ) M is a point on arc
BC. Lines AM, BM and CM are drawn.
Then ( overline{boldsymbol{A} boldsymbol{M}} ) is:
A. equal to ( overline{B M}+overline{C M} )
B. Less than ( overline{B M}+overline{C M} )
c. greater than ( overline{B M}+overline{C M} )
D. equal, less than, or greater than ( overline{B M}+overline{C M} ) depending upon the pos
9
254 ( O ) is the centre of the circle. If chord ( A B )
chord ( C D, ) then value of ( x ) is equal to
4.70
3.50
( c cdot 55 )
) . 45
9
255 In the figure, ( angle A C B=90^{0} ) and radius
of big circle ( =2 c m, ) then the radius of
small circle is (in ( c m) )
( A cdot 3-2 sqrt{2} )
B. ( 4-2 sqrt{2} )
c. ( 7-4 sqrt{2} )
D. ( 6-4 sqrt{2} )
9
256 67. The radius of two concentric cir-
cles are 9 cm and 15 cm. If the
chord of the greater circle be a
tangent to the smaller circle,
then the length of that chord is
(1) 24 cm (2) 12 cm
(3) 30 cm (4) 18 cm
9
257 The length of the chord of the circle ( (x-3)^{2}+(y-5)^{2}=80 ) cut off by the
line ( 3 x-4 y-9=0 ) is
A . 16
B. 8
( c cdot sqrt{96} )
( mathbf{D} cdot 2 sqrt{96} )
9
258 In the figure, ‘O’ is the centre of the
circle and ( 0 mathrm{M}, ) On are the
perpendiculars from the centre to the
chords ( P Q ) and ( R S . ) If ( O M=O N ) and ( P Q=6 )
( mathrm{cm} . ) Find RS
9
259 Consider a circle of radius ( R ). What is
the length of a chord which subtends an
angle ( theta ) at the centre?
( ^{mathrm{A}} cdot_{2 R sin frac{theta}{2}} )
B. ( 2 R sin theta )
c. ( _{2 R tan frac{theta}{2}} )
D. ( 2 R tan theta )
9
260 70. AB is a diameter of a circle with
centre at O. DC is a chord of it
such that DC | AB. If ZBAC =
20°, then 2 ADC is equal to
(1) 120 (2) 110°
(3) 115 (4) 100°
9
261 What is the volume in cubic cm of a
pyramid whose area of the base is
25 sq cm height ( 9 c m ? )
A ( cdot 75 mathrm{cm}^{3} )
B. ( 70 mathrm{cm}^{3} )
( mathrm{c} cdot 100 mathrm{cm}^{3} )
D. None of these
9
262 The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord? 9
263 Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of
the arc.
10
264 58. Each of the circles of equal radi
with centres A and B pass
through the centre of one anoth-
er circle they cut at C and D then
ZDBC is equal to
(1) 60°
(2) 100°
(3) 120° (4) 140°
9
265 The length of the shortest chord of the circles ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{2} boldsymbol{g} boldsymbol{x}+boldsymbol{2} boldsymbol{f} boldsymbol{y}+boldsymbol{c}=boldsymbol{0} )
which passes through the point ( (a, b) ) inside the circle
9
266 Recall that two circles are congruent if they have the same radii. Prove that
equal chords of congruent circles subtend equal angles at their centres.
9
267 A circle has the equation ( (x+1)^{2}+ )
( (y-3)^{2}=16 ) What are the coordinates
of its center and the length of its
radius?
A. (-1,3) and 4
B. (1,-3) and 4
c. (-1,3) and 16
D. (1,-3) and 16
9
268 Line segment joining the centre to any point on the circle is a radius of the circle.
A. True
B. False
9
269 Given inside a circle, whose radius is
equal to ( 13 mathrm{cm}, ) is a point ( mathrm{M} ) at a distance ( 5 mathrm{cm} ) from the centre of the
circle. A chord ( A B=25 mathrm{cm} ) is drawn
through M. The lengths of the segments into which the chord ( A B ) is divided by
the point ( M ) in ( C M ) are
A. 12,13
в. 14,11
c. 15,10
D. 16, 9
9
270 From an external point ( P, ) two tangents
PA and ( P B ) are drawn to the circle with
center ( 0 . ) Prove that OP is the
perpendicular bisector of ( A B )
10
271 Find ( Q M )
( A cdot 13 c m )
3. ( 12 mathrm{cm} )
( c .5 mathrm{cm} )
D. ( 8 mathrm{cm} )
9
272 Find equation of circle which passes through the point (2,3) and touches the line ( 2 x-3 y-13=0 ) at the point
(2,-3)
10
273 Consider a circle ( x^{2}+y^{2}+a x+b y+ )
( c=0 ) lying completely in first quadrant.
If ( mathrm{m}_{1} ) and ( mathrm{m}_{2} ) are the maximum and
minimum values of y/x for all ordered pairs ( (x, y) ) on the circumference of the
circle, then the value of ( left(boldsymbol{m}_{1}+boldsymbol{m}_{2}right) ) is
A ( cdot frac{a^{2}-4 c}{b^{2}-4 c} )
в. ( frac{2 a b}{b^{2}-4 c} )
c. ( frac{2 a b}{4 c-b^{2}} )
D. ( frac{2 a b}{b^{2}-4 a c} )
10
274 The length of tangent drawn from an
external point ( P ) to a circle with centre
0, is ( 8 mathrm{cm} ). If the radius of the circle is 6
( mathrm{cm}, ) then the length of OP (in cm) is :
A ( 2 sqrt{7} )
7
B. ( 4 sqrt{7} )
( c cdot 10 )
D. 10.5
10
275 If ( boldsymbol{A}=(mathbf{5}, mathbf{8}), ) then area of ( triangle boldsymbol{A B D} ) in
square units is
A ( cdot frac{96 sqrt{5}}{89} )
в. ( frac{960 sqrt{5}}{89} )
c. ( frac{960 sqrt{5}}{sqrt{8} 9} )
D. None of these
9
276 If two tangents are drawn to a circle
circle from an external point, the
(i) they subtend equal angles at the centre
(ii) they are equally inclined to the segment,joining the centre to that
point.
10
277 Circles with centres ( A, B ) and ( C ) touch
each other externally. If ( boldsymbol{A B}= )
( mathbf{3} c boldsymbol{m}, boldsymbol{B} boldsymbol{C}=mathbf{3} boldsymbol{c m}, boldsymbol{C} boldsymbol{A}=mathbf{4} boldsymbol{c m}, ) then
find the radii of each circle
9
278 A chord of a circle of radius ( 15 mathrm{cm} )
subtends an angle of ( 120^{circ} ) at the centre. Find the area corresponding minor sector of the circle.
9
279 69. In a cyclic quadrilateral ABCD, if
ZB-ZD = 60° then the measure
of the smaller of the two is :
(1) 60°
(2) 40°
(3) 38°
(4) 30°
9
280 The locus of the feet of perpendiculars drawn from the point ( (a, 0) ) on tangents
to the circle ( x^{2}+y^{2}=a^{2} ) is
A ( cdot a^{2}left(x^{2}+y^{2}+a xright)^{2}=a^{2}left(y^{2}+(x+a)^{2}right) )
B ( cdot a^{2}left(x^{2}+y^{2}-a xright)^{2}=y^{2}+(x-a)^{2} )
C ( cdotleft(x^{2}+y^{2}-a xright)^{2}=a^{2}left(y^{2}+(x-a)^{2}right) )
D cdot a ( ^{2}left[left(x^{2}+y^{2}right)-a^{2} x^{2}right]=left(y^{2}+(x-a)^{2}right) )
10
281 A circle passes through (0,0) and (1,0) and touches the circle ( x^{2}+y^{2}=9 )
then the centre of circle is
( ^{mathbf{A}} cdotleft[frac{3}{2}, frac{1}{2}right] )
В. ( left[frac{1}{2}, frac{3}{2}right] )
c. ( left[frac{1}{2}, frac{1}{2}right] )
D. ( left[frac{1}{2}, pm sqrt{2}right] )
9
282 f the length of the chord ( Y Z ) is equal to
the radius of the circle ( O Y ), find ( angle Y X Z )
A ( cdot 60^{circ} )
B. ( 30^{circ} )
( c cdot 80^{circ} )
D. 100
9
283 Two circles of radii ( 10 mathrm{cm} ) and ( 8 mathrm{cm} )
intersect each other and the length of the common chord is 12 m. Then the
distance between their centres is
( mathbf{A} cdot(10+2 sqrt{7}) mathrm{cm} )
B. ( (8+2 sqrt{7}) mathrm{cm} )
( mathbf{c} cdot(12+2 sqrt{7}) mathrm{cm} )
D ( cdot(6+2 sqrt{7}) ) ст
9
284 A chord ( A B ) is at a distance of ( 6 mathrm{cm} )
from the centre of a circle whose radius
is ( 6 mathrm{cm} ) less than that of the chord ( A B )
Then the length of the chord ( A B ) is
( A cdot 8 mathrm{cm} )
B. ( 32 mathrm{cm} )
c. ( 24 mathrm{cm} )
D. ( 16 mathrm{cm} )
9
285 If a chord of a circle ( x^{2}+y^{2}=32 )
makes equal intercepts of length ( l ) on
the co-ordinate axes, then
( mathbf{A} cdot ell<8 )
в. ( ell8 )
D. ( ell>16 )
9
286 Prove that the line joining the mid-
points of two parallel chords of a circle
passes through the centre.
9
287 Draw two tangents from a point ( 5 mathrm{cm} ) away from the centre of a circle of
radius ( 3 mathrm{cm} )
10
288 Find the value of ( x ) in each of the
following diagrams
( (mathbf{i}) )
(ii)
9
289 If radius of circle is ( 5 mathrm{cm} ) and distance
from centre to the point of intersection of 2 tangents in ( 13 mathrm{cm} . ) Find length of
tangent.
A . ( 11 mathrm{cm} )
B. ( 10 mathrm{cm} )
c. ( 12 mathrm{cm} )
D. ( 13 mathrm{cm} )
10
290 Equation of chord ( mathrm{AB} ) of circle ( x^{2}+ )
( boldsymbol{y}^{2}=boldsymbol{2} ) passing through (2,2) such
that ( boldsymbol{P B} / boldsymbol{P A}=mathbf{3}, ) is given by
A ( . x=3 y )
В. ( y-2=sqrt{3}(x-2) )
c. ( x=y )
D. none of these
9
291 If two equal chords of a circle intersect
within the circle, prove that the chords and line joining the point of intersection to the centre makes angles which are
A. Complementary to each other
B. Suplimentary to each other
c. Equal to each other
D. Not equal to each other
9
292 An equilateral triangle is inscribed in a circle of radius ( 6 mathrm{cm} . ) Find its side. 9
293 The radius of the circle ( x^{2}+y^{2}+x+ )
( c=0 ) passing through the origin is
A ( cdot frac{1}{4} )
в. ( frac{1}{2} )
c. 1
D. 2
9
294 A rectangle ABCD is inscribed in a circle
with centre 0. If AC is the diagonal and
( angle B A C=30^{circ}, ) then radius of the circle
will be equal to
A ( cdot frac{sqrt{3}}{2} B C )
B. BC
( c cdot sqrt{3} B C )
D. 2BC
9
295 65. In the given figure, PAB is a se-
cant and PT is a tangent to the
circle from P. If PT = 5 cm, PA =
4 cm and AB = x cm, then x is
5 cm
P
4 cm
A
x cm 7В
x cm
cm
(3) 5 cm
(43
cm
10
296 A steel wire, when bent in the form of a
square, encloses an area of 121 sq. ( mathrm{cm} )
The same wire is bent in the form of a
circle. Find the area of the circle.
9
297 From the figure, identify a point in the
exterior.
9
298 n Figure, ( P Q ) is a chord of length ( 8 mathrm{cm} )
of a circle of radius ( 5 mathrm{cm} . ) The tangents
at ( P ) and ( Q ) intersect at a point ( T . ) The
ength of ( T P ) is equal to ( frac{w}{3}, ) then the
value of ( a ) is
10
299 In the given figure, find the value of ( x )
( A cdot 25 )
3.30
( c .35 )
2.4
10
300 Two tangents are drawn to a circle from
an external point ( A ), touching the circle at the points ( P ) and ( Q . A ) third tangent
intersects segment ( boldsymbol{A P} ) at ( boldsymbol{B} ) and
segment ( A Q ) at ( C ) and touches the
circle at ( R ) If ( A Q=10 ) units, then the
perimeter of ( Delta A B C ) is
A . 22.0
в. 20.5
( c .20 .0 )
D. 40.0
10
301 Fill in the blanks
The longest chord of a circle is a
of the circle.
9
302 Area of circle in which a chord of length
( 2 sqrt{3} ) units, subtends angle ( 120^{circ} ) at its
centre is :
A . ( pi ) sq units
B. 2 ( pi ) sq units
c. ( 4 pi ) sq units
D. ( 5 pi ) sq units
9
303 Let ( S=x^{2}+y^{2}+2 g x+2 f y+c=0 )
be a given circle. Then the locus of the foot of the perpendicular drawn from the origin upon any chords of ( S ) which
subtends right angle at the origin is:
A ( cdot x^{2}+y^{2}+g x+f y+c / 2=0 )
B . ( x^{2}+y^{2}=g )
c. ( x^{2}+y^{2}=f )
D. ( x^{2}+y^{2}+g=0 )
9
304 Two circles of radii ( 20 mathrm{cm} ) and ( 37 mathrm{cm} )
intersect in ( A ) and ( B . ) If ( O_{1} ) and ( O_{2} ) are
their centres and ( A B=24 mathrm{cm}, ) then the
distance ( O_{1} O_{2} ) is equal to
A . ( 44 mathrm{cm} )
B. ( 51 mathrm{cm} )
( c .40 .5 mathrm{cm} )
D. ( 45 mathrm{cm} )
9
305 ( A B ) and ( C D ) are two parallel chords of a circle such that ( A B=10 mathrm{cm}, C D=24 mathrm{cm} )
If the chords are on opposite sides of the centre and distance between them is 17
( mathrm{cm}, ) the radius of the circle is
( A cdot 10 mathrm{cm} )
B. ( 11 mathrm{cm} )
c. ( 12 mathrm{cm} )
D. ( 13 mathrm{cm} )
9
306 In the following figure, ray PA is
tangent to the circle at ( A ) and ( P B C ) is a
secant. If ( A P=15, B P=10, ) then find ( B C )
10
307 If two equal chords of a circle intersect
each other, then prove that the
segments of one chord are equal to
corresponding segment of the other
chord.
9
308 In the figure on your right, 0 is the centre of the circle State
Which of the line segment are chords?
9
309 ( mathrm{M} ) and ( mathrm{N} ) are the mid-points of two equal
chords ( A B ) and ( C D ) respectively of ( a ) circle with centre ( 0 . ) Prove that
( angle A M N=angle C N M )
9
310 ( O ) is the centre of the circle having
radius ( 5 mathrm{cm} . O M ) is a ( perp ) on chord ( A B ). If
( O M=4 mathrm{cm}, ) then the length of the
chord ( A B ) is equal to
( A cdot 5 mathrm{cm} )
B. ( 6 mathrm{cm} )
( c cdot 8 mathrm{cm} )
D. ( 10 mathrm{cm} )
9
311 The longest chord of a circle is a ( ldots ). Of the circle.
A. Diameter
B. Lies on upper part of centre
c. Lies on lower part of centre
D. None of these
9
312 Given point are ( boldsymbol{P}=(mathbf{1},-mathbf{2}), boldsymbol{Q}=(mathbf{7}, mathbf{6}) )
is the origin. The length of the common chord of the circles with ( mathrm{OP} ) and ( mathrm{OQ} ) as diameters is
A . 1
B. 2
( c cdot 4 )
D. 6
9
313 Two circles of radii ( 10 mathrm{cm} ) and ( 8 mathrm{cm} )
intersects each other and the length of
the common chord is ( 12 mathrm{cm} ), find the
distance between their centers.
A. ( 2 mathrm{cm} )
в. ( (8+2 sqrt{7}) ) ст
( c .8 mathrm{cm} )
D. ( 2 sqrt{7} ) cm
9
314 A circle has two equal chords ( $ $ P Q $ $ ) and ( $ $ P R $ $ )
diameter ( $ $ P D $ $ ) cuts ( $ $ Q R $ $ ) in ( $ $ E )
( $ $ . ) If ( P R=12 c m ) and ( P E=8 c m, ) then
the length of ( $ $ ) PD ( $ $ ) is ( ? )
( mathbf{A} cdot 25 mathrm{cm} )
B . ( 22 mathrm{cm} )
c. ( 20 mathrm{cm} )
D. ( 18 mathrm{cm} )
9
315 Prove that if chords of congruent circles
subtend equal angles their centres, then the chords are equal.
9
316 Find the equation of a circle with centre
(2,2) and passes through the point (4,5)
9
317 ( angle A O B ) is
( mathbf{A} cdot 54^{circ} )
B. ( 72^{circ} )
( c cdot 90^{circ} )
( D cdot 108^{circ} )
10
318 69. PO and OR are two chords of a
circle and they are equally in-
clined to the diameter drawn
through Q. What is the relation
between PO and QR?
(1) PO 1 OR
(2) PO > QR
(3) PQ < OR
(4) PO = QR
10
319 ( A B ) and ( C D ) are two parallel chords of a circle such that ( A B=10 mathrm{cm} ) and ( mathrm{CD}=24 )
( mathrm{cm}, ) If the chords are on the opposite sides of the centre and the distance
between them is ( 17 mathrm{cm} ), the radius of the circle is:
A . ( 14 mathrm{cm} )
B. 10 ( mathrm{cm} )
( c cdot 13 mathrm{cm} )
D. ( 15 mathrm{cm} )
9
320 The line drawn from center of circle to
bisect a chord is perpendicular to the chord. Is this true? If true enter 1 else 0 .
9
321 59. Inscribed ZACB intercepts AB of
circle with centre 0. If the bisec-
tor of ZACB meets arc AB in M
then :
(1) m AM > m MB
(2) m AM<m MB
(3) m AM = m MB
(4) None of these
9
322 If the lines ( 3 x-4 y+4=0 ) and ( 6 x- )
( 8 y-7=0 ) are tangents to a circle,
then find the radius of the circle.
( A cdot 3 / 4 )
в. ( 4 / 3 )
( c cdot 1 / 4 )
D. ( 7 / 4 )
9
323 In the adjacent figure, ( A B ) is a chord of
circle with centre ( 0 . ) CD is the diameter
perpendicular to AB. Show that ( A D=B D )
9
324 Prove that the line joining a mid point of
a chord to the centre of circle is
perpendicular to it
9
325 Hilswer untulury quesuuris Na>tu
on the following.
( C_{1} ) and ( C_{2} ) are two circles and points
( boldsymbol{P}_{1}, boldsymbol{P}_{2}, boldsymbol{P}_{3}, boldsymbol{P}_{4}, boldsymbol{P}_{5} ) are noted. From which
point tangent is possible to ( C_{2} ) but not
( C_{1} )
( A cdot P_{2} )
в. ( P_{3} )
( c cdot P_{4} )
D. ( P_{5} )
10
326 Find the centre and radius of the circle
( boldsymbol{x}^{2}+boldsymbol{y}^{2}+2 boldsymbol{a} boldsymbol{x}-boldsymbol{2} boldsymbol{b} boldsymbol{y}+boldsymbol{b}^{2}=mathbf{0} )
9
327 Prove that the lengths of the tangents drawn from an external point to a circle
are equal.
10
328 69. If O be the circumcentre of a tri-
angle PQR and Z QOR= 110°, 2
OPR = 25°, then the measure of
PRO is
(1) 65°
(2) 50°
(3) 55° (4) 60°
9
329 A chord of a circle is ( 12 mathrm{cm} ) which is at a distance of ( 8 mathrm{cm} ) from center. Find the length of the chord of the same circle
which is at a distance of ( 6 mathrm{cm} ) from the
centre
( A cdot 20 mathrm{cm} )
B. ( 24 mathrm{cm} )
( c .16 mathrm{cm} )
D. ( mathrm{cm} )
9
330 Draw a circle and mark a diameter. 9
331 The chord of a ( odot(0,5) ) touches ( odot(0,3) )
The length of the chord is
( mathbf{A} cdot mathbf{8} )
B. 6
( c cdot 7 )
D.
9
332 ( P ) is a point on the common chord ( R S ) produced by two intersecting circles.
( A B ) and ( C D ) are the chords of the
circles,they meet at ( P ) produced.Prove that ( boldsymbol{P A} times boldsymbol{P B}=boldsymbol{P C} times boldsymbol{P D} )
9
333 In the given figure, ( P A ) and ( P B ) are two
tangents drawn from an external point
( P ) to a circle with centre ( O ). Prove that
OP is the right bisector of line segment
( A B )
10
334 70. ABCD is a cyclic quadrilateral.
The side AB is extended to E in
such a way that BE=BC. If ZADC
= 70°, ZBAD = 95°, then ZDCE
is equal to
(1) 140°
(2) 120°
(3) 165 (4) 110°
9
335 Find the value of ( x )
A ( cdot 50^{circ} )
B ( .60^{circ} )
( c cdot 70^{circ} )
D. ( 80^{circ} )
10
336 Find the angle marked ( a )
4.77
8. 36
( c cdot 41^{circ} )
( mathbf{D} cdot 13^{circ} )
10
337 The inner circumference of a circular
track is ( 24 pi mathrm{m} ). The track is ( 2 mathrm{m} ) wide
from everywhere. The quantity of wire required to surround the path completely is
A. ( 80 mathrm{m} )
B. ( 81 mathrm{m} )
c. ( 82 mathrm{m} )
D. 88m
9
338 In the following figure, ( Delta A B C ) is an isosceles triangles with perimeter
( 40 mathrm{cm} . ) The base ( A C ) is of length ( 10 mathrm{cm} )
Side ( A B ) and side ( B C ) are congruent. ( A )
circle touches the three sides as shown
in the figure below. Find the length of
the tangent segment from point ( B ) to
the circle.
10
339 n given figure ( C ) is centre of circle. ( A O )
and ( B O ) are tangents to circle. ( C M perp )
( A B )
( mathbf{f} boldsymbol{A} boldsymbol{C}=boldsymbol{A} boldsymbol{B}=mathbf{6} boldsymbol{c m}, ) then find ( boldsymbol{A} boldsymbol{M} )
( A cdot 3 c m )
8. 4 ст
( c .5 mathrm{cm} )
D. ( 6 mathrm{cm} )
9
340 Tangents are drawn from (4,4) to the ( operatorname{circle} x^{2}+y^{2}-2 x-2 y-7=0 )
meet the circle at ( A ) and ( B ). The length
of the chord ( A B ) is
A ( cdot 2 sqrt{3} )
3
B. ( 3 sqrt{2} )
c. ( 2 sqrt{6} )
D. ( 6 sqrt{2} )
9
341 Name the following part from the adjacent figure where ‘O’ is the center of
the circle.
( boldsymbol{A O} )
9
342 n the following figure, the line ABCD is
perpendicular to PQ ; where P and Q are the centres of the circles. Show that:
( A B=C D )
ii) ( A C=B D )
9
343 Prove that if chords of congruent circles subtend equal angles at their centre, then the chords are equal. 9
344 In two concentric circle, prove that all
chords of the outer circle which touch
the inner are of equal length.
9
345 Find the equation of the circle whose center lies on the positive direction of ( y ) axis at a distance 6 from the origin and
whose radius is 4
9
346 Prove that the angle in a semicircle is a
right angle.
9
347 67. O and C are respectively the or-
thocentre and circumcentre of an
acute-angled triangle PQR. The
points P and O are joined and
produced to meet the side QR at
S. If ZPQS = 60° and ZQCR =
130°, then ZRPS =
(1) 30° (2) 35°
(3) 100
(4) 60°
9
348 If ( operatorname{lines} x-2 y+3=0,3 x+k y+7= )
0 cut the coordinate axes in concyclic
points, then ( k=? )
( mathbf{A} cdot 3 / 2 )
B. ( 1 / 2 )
c. ( -3 / 2 )
D. -4
9
349 In the figure ( P Q ) is tangent to the circle
at ( p t )
Find the radius, if ( P Q=8 mathrm{cm} ) and
( boldsymbol{O} boldsymbol{R}=mathbf{1 0} boldsymbol{c m} )
10
350 Two parallel chords are drawn on the
same side of the centre of a circle of
radius 20. It is found that they subtend
( 60^{0} ) and ( 120^{0} ) angles at the centre of the circle. Then the perpendicular distance between the chords is:
( mathbf{A} cdot 5(sqrt{3}-1) )
B . ( 10(sqrt{3}-1) )
c. ( 10(sqrt{2}-1) )
() ( 5(sqrt{2}-1-1) )
D. ( 5(sqrt{3}+1) )
9
351 A line segment whose end points lie on the circle is called ( ldots ldots . . . . . . . ) to the circle.
A. Chord
B. tangent
c. Radius
D. Diameter
9
352 In a diagram ( boldsymbol{O} ) is the centre of circle.
Calculate the value ( a )
A . 43
B. 53
( c cdot 63 )
D. 33
9
353 In the figure, ‘O’ is the centre of the
circle. ( O M=3 mathrm{cm} ) and ( mathrm{AB}=8 mathrm{cm} ). Find
the radius of the circle.
A ( .5 mathrm{cm} )
B. 4 cm
c. ( 15 mathrm{cm} )
D. ( 8 c m )
9
354 n figure if PQR is tangent to circle at ( mathrm{Q} )
whose centre is ( 0 . A B ) is a chord parallel
to PR and ( angle B Q R=70^{circ} ) then ( angle A Q B ) is
equal to
( mathbf{A} cdot 20 )
B. 40
( c .35 )
D. 45
10
355 Establish the formula for area and
circumference of circle.
9
356 If ( L equiv 2 x+y-6=0, ) then the locus of
circumcentre of ( triangle P Q R ) is
A. ( 2 x-y=4 )
в. ( 2 x+y=3 )
c. ( x-2 y=4 )
D. ( x+2 y=3 )
10
357 A tangent ( mathrm{PQ} ) at a point ( mathrm{P} ) of a circle of radius ( 5 mathrm{cm} ) meets a line through the centre 0 at a point ( Q, ) so that ( O Q=12 )
cm. Length of PQ is :
A. ( sqrt{112} mathrm{cm} )
B . ( sqrt{113} mathrm{cm} )
c. ( sqrt{85} mathrm{cm} )
D. ( sqrt{119} mathrm{cm} )
10
358 55. AB is the diameter of circle and
AC is its one chord. The tangent
at C intersect the produced di-
ameter AB at D. Given that AB =
10 cm, AC = 8 cm ZBAC = 30°
then BD will be equal to
(1) 6 cm (2) 8 cm
(3) 10 cm (4) 4 cm
9
359 If figure ( C E ) and ( D E ) are equal chords
of a circle with centre ( O ). If ( angle A O B= )
( 90^{circ}, ) find ratio of the area of ( triangle C E D ) and
( triangle A O B )
9
360 ( boldsymbol{S} boldsymbol{R}=? )
( 4 . overline{P Q} )
3. ( overline{P Q} )
: Q
( . overline{S R} )
9
361 The length of a tangent from a point ( boldsymbol{A} )
at distance ( 5 mathrm{cm} ) from the centre of the
circle is ( 4 mathrm{cm} ). Find the radius of the
circle.
10
362 A chord of a circle divides the circular
region in two parts the region which contains the centre is known as
A. minor Arc
B. major Arc
c. minor Segment
D. major Segment
9
363 Draw a pair of tangents to a circle of
radius ( 3 mathrm{cm} ) which are inclined to each
other at an angle of ( 45^{circ} )
10
364 Coordinates of the centre of the circle
which bisects the circumferences of the
circles ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=mathbf{1}: boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{2} boldsymbol{x}- )
( mathbf{3}=mathbf{0} ) and ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+mathbf{2} boldsymbol{y}-mathbf{3}=mathbf{0} ) is
A ( cdot(-3,-3) )
B. (3,3)
c. (2,2)
D. (-2,-2)
9
365 Two circles with centres ( A ) and ( B ) of
radii ( 3 mathrm{cm} ) and ( 4 mathrm{cm}, ) respectively
intersect at two points ( C ) and ( D ) such that ( A C ) and ( B C ) are tangents to the
two circles. Find the 10 times length of
the common chord ( C D )
A .48
B. 58
( c cdot 56 )
D. 54
9
366 What is a line passing through two points on a circle called?
A. secant
B. Digonal
c. Radius
D. tangent
10
367 The common chord of the circles ( x^{2}+ )
( boldsymbol{y}^{2}-mathbf{4} boldsymbol{x}-mathbf{4} boldsymbol{y}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}^{2}+mathbf{2} boldsymbol{y}^{2}=mathbf{3} mathbf{2} )
subtends at the origin an angle equal to
A ( cdot frac{pi}{3} )
B.
c.
D.
9
368 In the above figure, 0 is the centre of the
circle. The angle ( C B D ) is equal to
( A cdot 25 )
B. 50
( c cdot 40^{circ} )
D. 130
9
369 Tangents are drawn from the point ( (a, a) ) to the circle ( x^{2}+y^{2}-2 x-2 y- )
( 6=0 . ) If the angle between the tangents lies in the range ( left(frac{pi}{3}, piright), ) then the exhaustive range of values of ( a ) is
B. (-5,-3)( cup(3,5) )
c. ( (-infty, 2 sqrt{2}) cup(2 sqrt{2}, infty) )
D. (-3,-1)( cup(3,5) )
10
370 If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to
corresponding segments of the other chord.
9
371 65. The chord of a circle is equal to
its radius. The angle subtended
by this chord at the minor arc of
the circle is
(1) 75° (2) 60°
(3) 150 (4) 120°
9
372 n fig, ( O ) is the centre of a circle ( A B= )
( mathbf{1 6} mathrm{cm}, boldsymbol{C D}=mathbf{1 4} mathrm{cm}, operatorname{seg} boldsymbol{O} boldsymbol{M} perp mathbf{s e g} )
( A B, operatorname{seg} O N perp operatorname{seg} C D . ) If ( O M=6 mathrm{cm} )
then length of ( operatorname{seg} O N ) is ( sqrt{m} mathrm{cm} . ) So, ( m )
is
A ( . m=149 c m^{2} )
В ( cdot m=51 c m^{2} )
( mathrm{c} cdot m=51 mathrm{cm} )
D. ( m=149 mathrm{cm} )
9
373 OA.OB are the radii of a circle with ( O )
as centre, the angle ( A O B=120^{circ} )

Tangents at ( A ) and ( B ) are drawn to meet
in the point ( C . ) If ( O C ) intersects the circle in the point ( D ), then ( D ) divides
( O C ) in the ratio
A . 1: 2
B. 1: 3
c. 1: 1
D. 2: 3

10
374 In a circle of radius ( 25 mathrm{cm} ) two parallel chords of the length ( 14 mathrm{cm} ) and ( 48 mathrm{cm} ) respectively, are drawn on the same
side of the centre. The distance between
them is
A . ( 14 mathrm{cm} )
B. ( 24 mathrm{cm} )
( c cdot 17 mathrm{cm} )
D. ( 31 mathrm{cm} )
9
375 f two equal chords of a circle intersect
within the circle, prove that the line
joining the point of intersection to the centre makes equal angles with the
chords.
9
376 The moon’s distance from the earth is
( 360000 mathrm{km} ) and its diameter subtends
an angle of ( 42^{prime} ) at the eye of the observer. The diameter of the moon is?
A. ( 4400 mathrm{km} )
B . ( 1000 mathrm{km} )
( mathbf{c} .3600 mathrm{km} )
D. ( 8800 mathrm{km} )
9
377 Find the radius of a circle whose
diameter has endpoints (-3,-2) and ( (7, )
8)
A. 5
B. ( 5 sqrt{2} )
( c cdot(2,3) )
D. ( sqrt{52} )
E. none of these
9
378 Consider the following diagram where
( A B ) and ( C D ) are congruent arcs and
chords. The measure of ( angle A O B=50^{circ} )
Then the value of ( angle C O D=? )
A ( .45^{circ} )
В. ( 50^{circ} )
( c cdot 56^{circ} )
D. ( 90^{circ} )
9
379 Perimeter of a circle is called its
A . circumference
B. area
c. diameter
D. none of these
9
380 n the figure if ( angle B D C=30^{circ}, angle )
( C B A=110^{circ}, ) then find ( angle B C A )
4.20
3.40
235
; 0
9
381 ( A, B, C ) are three points on a circle such
that ( A B ) is the chord and ( C P ) is the
perpendicular to ( O P, ) where ( O ) is the
centre and ( P ) is any point on ( A B . ) The
radius ( r ) of the circle is given by
A ( cdot r^{2}=O P^{2}+A P times C P )
B . ( r^{2}=O P^{2}+A P times P B )
c. ( r^{2}=O P^{2}+P B times P C )
D . ( r^{2}=O P^{2}+P B^{2} )
9
382 Calculate the length of a chord which is at a distance of ( 12 mathrm{cm} ) from the centre
of a circle of radius ( 13 mathrm{cm} )
9
383 In the given figure, ( P Q mathrm{cm}, M ) is the
mid-point of ( boldsymbol{Q} boldsymbol{R} ) ?
Also, ( M N perp P R, Q S=7 mathrm{cm} ) and ( T R= )
( 21 c m, ) then ( M N=? )
( mathbf{A} cdot 14 mathrm{cm} )
( mathbf{B} cdot 12.5 mathrm{cm} )
c. ( 31 mathrm{cm} )
D. 25 cm
10
384 Distance of chord ( A B ) from the centre of
a circle is ( 8 mathrm{cm} ). Length of the chord ( A B )
is ( 12 mathrm{cm} . ) Find the diameter of the circle
9
385 52. Two circles of diameters 10 cm
and 6 cm have the same cen-
tre. A chord of the larger circle
is a tangent of the smaller
one. The length of the chord is
(1) 4 cm. (2) 8 cm.
(3) 6 cm. (4) 10 cm.
9
386 If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of other chord 9
387 71. The tangents drawn at P and Q
on the circumference of a circle
intersect at A. If Z PAQ = 68°,
then the measure of the Z APO
(1) 56°
(3) 28°
(2) 680
(4) 34°
10
388 The length of the chord of a circle is ( 8 mathrm{cm} ) and perpendicular distance between centre and the chord is ( 3 mathrm{cm} ). Then the
radius of the circle is equal to?
A. ( 4 mathrm{cm} )
B. ( 5 mathrm{cm} )
( mathrm{c} cdot 6 mathrm{cm} )
D. ( 8 mathrm{cm} )
9
389 The angle between the two tangents from the origin to the circle ( (x-7)^{2}+ )
( (y+1)^{2}=25 ) equals-
( ^{A} cdot frac{pi}{2} )
в.
( c cdot frac{pi}{4} )
D. None of these.
10
390 In the given figure given below ( P Q ) is a
diameter chord ( S R ) is parallel to ( P Q )
Given ( angle P Q R=58^{circ}, ) calculate ( angle R P Q )
( A cdot 30^{circ} )
B. 32
( c cdot 34 )
( D .36 )
9
391 Draw any circle and mark a sector 9
392 Prove that the line joining the mid-
points of two equal chords of a circle subsent equal angles with the chord.
9
393 71.
ord PQ is
2 is the per
PO at M
In a given circle, the chord Po
of length 18 cm. AB is the
pendicular bisector of PQ af
If MB = 3 cm, then the length
AB is
LA
un of
79
(1) 27 cm.
(3) 28 cm.
(2) 30 cm.
(4) 25 cm.
9
394 Tangents PA and PB drawn to ( x^{2}+y^{2}= )
9 from any arbitrary point ‘P’ on the line ( x+y=25 . ) Locus of midpoint of chord
( A B ) is
A ( cdot 25left(x^{2}+y^{2}right)=9(x+y) )
B . ( 25left(x^{2}+y^{2}right)=3(x+y) )
C. ( 5left(x^{2}+y^{2}right)=3(x+y) )
D. None of these
10
395 In the figure given, ( O ) is the centre of the
circle. ( A B ) and ( C D ) are two chords of the
circle. ( O M ) is perpendicular to ( A B ) and ( O N ) is perpendicular to ( C D . A B= )
( mathbf{2 4} c boldsymbol{m}, boldsymbol{O} boldsymbol{M}=mathbf{5} boldsymbol{c m}, boldsymbol{O} boldsymbol{N}=mathbf{1 2} c boldsymbol{m} . ) Finc
the
Length of chord ( C D )
9
396 Find the centre and radius of the circle
( x^{2}+y^{2}+6 x+8 y-96=0 )
9
397 A chord of length ( 30 mathrm{cm} ) is drawn at a
distance of ( 8 mathrm{cm} ) from the centre of
a circle. The radius of the circle (in cm.)
is
( mathbf{A} cdot 15 mathrm{cm} )
B. ( 21 mathrm{cm} )
c. ( 18 mathrm{cm} )
D. ( 17 mathrm{cm} )
9
398 If two circles intersect at two points, prove that their centres lie on the
perpendicular bisector of the common chord
9
399 Prove that the length of the common chord of the two circles whose
equations are ( (x-a)^{2}+(y-b)^{2}=c^{2} )
and ( (x-b)^{2}+(y-a)^{2}=c^{2} ) is
( sqrt{4 c^{2}-2(a-b)^{2}} )
Hence find the condition that the two
circles may touch.
9
400 In the figure, the chord BD is
perpendicular to the diameter AC. Find
the measures of the following angles.
a. ( angle B A C )
b. ( angle B C D )
c. ( angle boldsymbol{A} boldsymbol{D} boldsymbol{C} )
( mathrm{d} . angle C D M )
e. ( angle B A P )
9
401 If two chords of lengths ( 2 a ) each, of a
circle of radius ( R, ) intersect each other
at right angles then the distance of their point of intersection from the
centre of the circle is
A ( cdot 2 sqrt{R^{2}-a^{2}} )
the ( sqrt{R^{2}-a^{2}} )
B . ( sqrt{2left(R^{2}-a^{2}right)} )
c. ( 4 sqrt{left(R^{2}-a^{2}right)} )
D ( cdot 2left(R^{2}-a^{2}right) )
9
402 If tangents ( boldsymbol{T} boldsymbol{A} ) and ( boldsymbol{T} boldsymbol{B} ) from a point ( boldsymbol{T} )
to a circle with centre ( O ) are inclined to
each other at an angle of ( 70^{circ}, ) then find
( angle A O B ) (in degrees)
10
403 ( O ) is centre of the circle. Find the length
of radius, if the chord of length ( 24 mathrm{cm} ) is
at a distance of ( 9 mathrm{cm} ) from the centre of
the circle.
9
404 In the figure given above, ( A D ) is a
straight line, ( O P ) perpendicular to ( A D )
and 0 is the centre of both circles. If
( boldsymbol{O A}=mathbf{2 0} boldsymbol{c m}, boldsymbol{O B}=mathbf{1 5} boldsymbol{c m} ) and ( boldsymbol{O P}= )
( 12 mathrm{cm} . ) what is ( A B ) equal to?
( A cdot 7 mathrm{cm} )
( 3.8 mathrm{cm} )
( c .10 mathrm{cm} )
( 0.12 mathrm{cm} )
9
405 Prove that out of all the chords
which passing through any point circle, that chord will be smallest which is
perpendicular on diameter which passes through that point.
9
406 The
distance from the centre to the
circumference.
A. Sector
B. Segment
c. Diameters
D. Radius
9
407 ( A B ) and ( C D ) are two parallel chords of a circle such that ( A B=10 mathrm{cm} ) and
( C D=24 mathrm{cm} . ) If the chords are on
opposite sides of the centre and the distance between them is ( 17 mathrm{cm} . ) The
radius of the circle is
( mathbf{A} cdot 26 mathrm{cm} )
B. ( 39 mathrm{cm} )
c. ( 6.5 mathrm{cm} )
D. ( 13 mathrm{cm} )
9
408 67. If a chord of length 16 cm is at a
distance of 15 cm from the cen-
tre of the circle, then the length
of the chord of the same circle
which is at a distance of 8 cm
from the centre is equal to
(1) 10 cm (2) 20 cm
(3) 30 cm (4) 40 cm
9
409 Two parallel chords in a circle are ( 10 mathrm{cm} ) and ( 24 mathrm{cm} ) long. If the radius of the
circle is ( 13 mathrm{cm} ), find the distance between the chords if thay lie on the opposite sides of the center
9
410 Two equal circles in the same plane can have at the most the following numbers of common tangents
( A cdot 3 )
B . 2
( c cdot 4 )
D.
10
411 Which is a secant?
( A cdot ) ми
B. on
( c . P Q )
D. None
10
412 70. Two chords AB and CD of cri-
cle whose centre is O, meet at
the point P and 2 AOC = 50°
BOD = 40°. Then the value
of BPD is
(1) 60°
(2) 40°
(3) 45° (4) 75°
9
413 Find the area of the sector of a circle
whose radius is ( 14 mathrm{cm} ) and angle of
sector is ( 45^{circ} )
9
414 The tangents drawn from the origin to the circle ( x^{2}+y^{2}+2 g x+2 f y+f^{2}= )
0 are perpendicular, if
A. ( g=f )
в. ( g=2 f )
c. ( 2 g=f )
D. ( 3 g=f )
10
415 Find the coordinates of a point ( A, ) where
( A B ) is the diameter of a circle whose
centre is (2,-3) and ( B ) is (1,4)
9
416 If ( alpha ) is the angle subtended at ( Pleft(x_{1}, y_{1}right) )
by the circle ( S=x^{2}+y^{2}+2 g x+ )
( 2 f y+c=0, ) then
This question has multiple correct options
A ( cdot cot alpha=frac{sqrt{S_{1}}}{sqrt{g^{2}+f^{2}-c}} )
B. ( cot alpha / 2=frac{sqrt{S_{1}}}{sqrt{g^{2}+f^{2}-c}} )
( ^{mathrm{c}} tan alpha=frac{2 sqrt{g^{2}+f^{2}-c}}{sqrt{S_{1}^{1}}} )
D. ( quad alpha=2 tan ^{-1}left(frac{sqrt{g^{2}+f^{2}-c}}{sqrt{S_{1}}}right) )
10
417 What are the coordinates of the center
of this circle?
( boldsymbol{x}^{2}+(boldsymbol{y}+mathbf{7})^{2}=mathbf{1 1} )
( A cdot(7,7) )
B. (0,7)
c. (-7,-7)
D. (0,-7)
9
418 If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to
corresponding segments of the other chord.
9
419 In the given figure, 0 is the centre of the
circle and ( angle A B C=36^{circ} . ) The measure
of ( angle A O C ) is :
( A cdot 36 )
B. 72
( c cdot 144 )
( D cdot 18 )
9
420 70. Two circles of radii Rand r touch
each other externally and PQ is
the direct common tangent, Then
PO2 is equal to:
(1) R-T (2) R+T
(3) 2R
(4) 4R
10
421 Find ( boldsymbol{P} boldsymbol{M} )
( mathbf{A} cdot 3 c m )
( mathbf{B} cdot 4 c m )
( mathbf{c} cdot 5 c m )
D. ( 8 mathrm{cm} )
9
422 ( O ) is the centre of the circle with radius
( 5 mathrm{cm} . ) Chords ( A B ) and ( C D ) are parallel.
( A B=6 mathrm{cm} ) and ( C D=8 mathrm{cm} . ) If ( P Q ) is
distance between ( A B ) and ( C D ), then the
length of ( boldsymbol{P Q} ) is
( mathbf{A} cdot 10 mathrm{cm} )
B. ( 8 mathrm{cm} )
( c cdot 7 mathrm{cm} )
D. ( 7 sqrt{2} mathrm{cm} )
9
423 The length of a minor arc is ( frac{2}{9} ) of the circumference of the circle. Write the
measure of the angle subtended by the arc at the centre of the circle.
9
424 Given ( B D=12 ) and ( A C=3 ) in the
circle with center ( A ). Find the radius.
A . 3
B. ( 3 sqrt{5} )
( c cdot 4 )
D. ( 4 sqrt{5} )
9
425 The common point of the tangent and circle is called
A. Intersecting points
B. Secant
c. Point of contact
D. None
10
426 The radius of a circle with centre 0 is 13
( mathrm{cm} . ) The distance of a chord from the
centre is ( 5 mathrm{cm} . ) Find the length of the chord.
( mathbf{A} cdot 24 mathrm{cm} )
B. ( 12 mathrm{cm} )
( mathrm{c} cdot 13 mathrm{cm} )
D. ( 26 mathrm{cm} )
9
427 n the figure, ( P Q=R S ) and ( angle O R S= )
( 48^{circ} ) Find ( angle O P Q ) and ( angle R O S )
9
428 The length of chord of circle with radius 10cm drawn at a distance of 8cm
( A cdot 12 mathrm{cm} )
B. ( 10 mathrm{cm} )
( c cdot 14 c m )
D. 30cm
9
429 In the given figure 0 is the centre of the
circle ( 0 B=5 c m, ) Distance from 0 to
Chord ( A B ) is ( 3 c m ).Find the length of ( A B )
9
430 From the following figure find value of
( boldsymbol{X} boldsymbol{Y} ) if ( angle boldsymbol{A} boldsymbol{O} boldsymbol{B}=angle boldsymbol{X} boldsymbol{O} boldsymbol{Y} )
3. 11
( r )
2
9
431 66. Each of the circles of equal radit
with centres A and B pass
through the centre of one anoth-
er circle they cut at C and D then
ZDBC is equal to
(1) 60°
(2) 100°
(3) 120
(4) 140°
9
432 Find the length of longest chord of the circle if radius is ( 2.9 mathrm{cm} ) in ( mathrm{cm} ) 9
433 At one end ( A ) of a diameter ( A B ) of a circle
of radius ( 5 mathrm{cm}, ) tangent ( mathrm{XAY} ) is drawn to the circle. Find the length of the chord
CD parallel to XY and at a distance ( 8 mathrm{cm} ) from A.
9
434 Two equal chords of a circle intersect
within the circle Then the
corresponding segments of the chords
are
A . not always equal
B. not equal
c. not related anyway
D. equal
9
435 In the diagram, 0 is the centre of the
circle. The angles CBD is equal to
( A cdot 120 )
B. ( 55^{circ} )
( c cdot 65 )
D. 75
9
436 ( A B ) is a chord of a circle with center 0
The tangent at B cuts AO produced at T
if ( angle B A T=25^{circ} ) Then the value of
( angle B T A ) is
A . ( 30^{circ} )
B. ( 60^{circ} )
( c cdot 25 )
D. ( 40^{circ} )
9
437 The lengths of the two tangents from an external point to a circle are
A. equal
B. different
c. both A and B
D. none of the above
10
438 In the given figure, ( boldsymbol{T} boldsymbol{T}^{prime} ) is the tangent
line. Which one of the following
relationship is true?
( mathbf{A} cdot x+y=2 z )
B . ( x+y=z )
c. ( z-3 x=y )
D. ( z-2 x=y )
10
439 The value of ( c, ) for which the line ( y= )
( 2 x+c ) is a tangent to the circle ( x^{2}+ )
( boldsymbol{y}^{2}=mathbf{1 6}, ) is
A . ( -16 sqrt{5} )
B. 20
c. ( 4 sqrt{5} )
D. ( 16 sqrt{5} )
10
440 The equation to the sides ( A B, B C, C A )
of a ( triangle operatorname{are} boldsymbol{x}+boldsymbol{y}=mathbf{1} ; mathbf{4} boldsymbol{x}-boldsymbol{y}+mathbf{4}= ) and
( 2 x+3 y=6 . ) Circle are drawn on
( A B, B C, C A ) as diameter. The point of
concurrence of the common chord is
A. centroid of the triangle
B. orthocenter
c. circumcenter
D. incenter
9
441 In the given figure, 0 is the centre of the
circle. If ( angle A O D=140^{circ} ) and ( angle C A B= )
( 50^{circ}, ) Then
(i) ( angle boldsymbol{E} boldsymbol{D} boldsymbol{B} ) (ii) ( angle boldsymbol{E} boldsymbol{B} boldsymbol{D} ) are
respectively
( begin{array}{lll}A & -70^{circ} & & 50end{array} )
( begin{array}{lll}text { B } cdot 50^{circ} & 110^{circ}end{array} )
( begin{array}{lll}c cdot 30^{circ} & & 70^{circ}end{array} )
( begin{array}{lll}text { D. } 120^{circ} & text { & } 130^{circ}end{array} )
9
442 Length of the common chord of the
( operatorname{circles}(x-1)^{2}+(y+1)^{2}= )
( c^{2} ) and ( (x+1)^{2}+(y-1)^{2}=c^{2} ) is
A ( cdot frac{1}{2} sqrt{c^{2}-2} )
B. ( sqrt{c^{2}-2} )
c. ( 2 sqrt{c^{2}-2} )
D. ( (c+2) )
9
443 A line touches a circle of radius ( 4 mathrm{cm} )
Another line is drawn which is tangent
to the circle. If the two lines are parallel then distance between them is
A ( .4 mathrm{cm} )
в. 6 ст
( c .7 c m )
D. ( 8 mathrm{cm} )
10
444 If the line ( y-m x+m-1=0 ) cuts the
( operatorname{circle} x^{2}+y^{2}-4 x-4 y+4=0 ) at two
real points, then ( m ) belongs to
A . [1,1]
B . [-2,2]
( c cdot(-infty, infty) )
D. [-4,4]
9
445 The equation of the diameter of circle ( x^{2}+y^{2}+2 x-4 y-11=0 ) which
bisects the chords intercepted on the line ( 2 x-y+3=0 ) is
A. ( x+y-7=0 )
В. ( 2 x-y-5=0 )
c. ( x+2 y-3=0 )
D. None of these
10
446 If ( omega ) is a cube root of unity, then ( (3+ ) ( left.mathbf{5} boldsymbol{omega}+mathbf{3} boldsymbol{omega}^{2}right)^{2}+left(mathbf{3}+mathbf{3} boldsymbol{omega}+mathbf{5} boldsymbol{omega}^{2}right)^{2} ) is equal
to
A . 4
B.
( c cdot-4 )
D. None of these
9
447 8.
In a triangle ABC, let C = Ifr is the inradius and R is
Toumradius of the triangle ABC, then 2 (r+R) equals
[2005]
(a) b+c (b) a + b (c) a+b+c (d) cta
atte
9
448 In which circles, angles at the centers make a equal chords?
A. concentric circles
B. eccentric circles
c. tangential circles
D. equal circles
9
449 A chord of length ( 16 mathrm{cm} ) is drawn in a
circle of radius ( 10 mathrm{cm} . ) The distance of
the chord from the centre of the circle is
( A cdot 8 mathrm{cm} )
B. ( 12 mathrm{cm} )
( c cdot 6 c m )
D. ( 10 mathrm{cm} )
9
450 Three wires of length ( l_{1}, l_{2}, l_{3} ) form a
triangle surmounted by another
circular wire, If ( l_{3} ) is the diameter and
( l_{3}=2 l_{1}, ) then the angle between ( l_{1} ) and
( l_{3} ) will be
A ( .30^{circ} )
B. ( 60^{circ} )
( c cdot 45^{circ} )
D. ( 90^{circ} )
9
451 The distance, once around the circle is
called
A. diameter
B. center
c. circumference
D. chord
9
452 Two chords of lengths ( 30 mathrm{cm} ) and ( 16 mathrm{cm} ) are on the opposite side of the centre of the circle. If the radius of the circle is 17
( mathrm{cm}, ) find the distance between the
chords.
9
453 ( A B ) and ( C D ) are two parallel chords of a circle such that ( A B=10 mathrm{cm} ) and
( C D=24 mathrm{cm} . ) If the chords are on the
opposite sides of the centre and the distance between them is ( 17 mathrm{cm} ), the radius of the circle is
A . ( 14 mathrm{cm} )
B. ( 10 mathrm{cm} )
c. ( 13 mathrm{cm} )
D. ( 15 mathrm{cm} )
9
454 The line ( y=x ) is a tangent at (0,0) to a
circle of radius is ( 1, ) then centre of the
circle is
( ^{mathbf{A}} cdotleft(frac{1}{sqrt{2}}, frac{1}{sqrt{2}}right) )
B ( cdotleft(frac{1}{2 sqrt{2}}-frac{1}{sqrt{2}}right) )
( ^{mathbf{c}} cdotleft(frac{-1}{sqrt{2}}, frac{1}{sqrt{2}}right) )
D. ( left(frac{-1}{sqrt{2}}, frac{-1}{sqrt{2}}right) )
10
455 ( A B ) is chord of a circle with centre ( O )
and radius ( 17 mathrm{cm} . ) If ( O M perp A B ) and
( boldsymbol{O} boldsymbol{M}=mathbf{8} mathrm{cm} . ) The length of chord ( boldsymbol{A B} ) is
A . ( 12 mathrm{cm} )
B. ( 30 mathrm{cm} )
c. ( 15 mathrm{cm} )
D. ( 24 mathrm{cm} )
9
456 In the figure, if ( boldsymbol{A B}=boldsymbol{C D} ) and
( angle A O B=90^{circ} ) find ( angle C O D )
9
457 In the given figure, ( O ) is the centre of a
circle. If ( A B ) and ( A C ) are chords of the
circle such that ( A B=A C ) and ( O P perp )
( A B, O Q perp A C, ) prove that ( P B=Q C )
9
458 Find the length of chord of circle with
radius ( 5 c m ) and distance from center
( 2 c m )
9
459 Find the value of ( x )
( mathbf{A} cdot x=10 )
B. ( x=8 )
( mathbf{c} cdot x=6 )
D. ( x=3 )
9
460 i) A circle can have ( _{–} ) -parallel tangents.
ii) The point common to the tangent and the circle is called
10
461 The circle and the square have the
same center and the same area. If the
circle has radius ( 1, ) the length of ( A B ) is
A ( .4-7 )
B . ( 4-2 sqrt{pi} )
( c cdot 2-sqrt{pi} )
D. ( sqrt{4-pi} )
9
462 n the given figure, 0 is the centre of the
circle. If ( angle B A D=75^{circ} ) and ( B C=C D )
find ( angle B O D, angle B C D, angle O B D )
9
463 Two tangents PT and PT’ are drawn to a
circle, with centre ( 0, ) from an external
point P. Prove that ( angle mathrm{TPT}^{prime}=2 angle mathrm{OTT} ) ‘.
10
464 23. A circle is drawn in a sector of a larger circle of radius r,
as shown in figure. The smaller circle is tangent to the
two bounding radii and the arc of the sector. The radius
of the smaller circle is
b.
nie
60°
10
465 Find the value of ( x ) and ( y )
A ( cdot x=10^{circ}, y=7 )
B . ( x=18^{circ}, y=5 )
( mathbf{C} cdot x=9^{o}, y=6 )
D. ( x=7^{circ}, y=6 )
9
466 The centre of a circle touching two intersecting lines lies on the angle bisector of the lines.
A. True
B. False
9
467 ( A D ) is a diameter of a circle and ( A B ) is
a chord. If ( boldsymbol{A} boldsymbol{D}=mathbf{3 4} boldsymbol{c m}, boldsymbol{A B}=mathbf{3 0} boldsymbol{c m} )
the distance of ( A B ) from the centre of
the circle is
( A cdot 17 mathrm{cm} )
( 3.15 mathrm{cm} )
( c .4 mathrm{cm} )
( 8.8 m )
9
468 If ( A B ) is tangent to the circle at ( A ) and
( O B=13 mathrm{cm}, ) find the radius ( O A )
( 4.5 mathrm{cm} )
( 3.7 mathrm{cm} )
( c cdot 8 c m )
)
10
469 If the diameter of circle is ( 10 mathrm{cm}, ) then find the radius of circle. 9
470 The condition that the chord ( x cos alpha+ )
( boldsymbol{y} sin boldsymbol{alpha}-boldsymbol{p}=mathbf{0} ) of ( boldsymbol{x}^{2}+boldsymbol{y}^{2}-boldsymbol{a}^{2}=mathbf{0} ) may
subtend a right angle at the centre of the circle is
A ( cdot a^{2}=2 p^{2} )
B ( cdot p^{2}=2 a^{2} )
c. ( a=2 p )
D. ( p=2 a )
9
471 In the figure, line ( A B ) is a tangent to
both the circles touching at ( A ) and ( B )
( boldsymbol{O} boldsymbol{A}=mathbf{2 9}, boldsymbol{B P}=mathbf{1 8}, ) and ( boldsymbol{O P}=boldsymbol{6 1 .} ) The
length of ( boldsymbol{A B} ) is
( A cdot 61 c m )
B. ( 60 mathrm{cm} )
c. ( 47 mathrm{cm} )
D. ( 11 c m )
10
472 Find the radius of that circle whose area
is ( 616 mathrm{cm}^{2}(text { in } mathrm{cm} .) )
9
473 67. Two circles touch internally at
a point P and form a point T
on the common tangent at P,
tangent segments TQ, TR are
drawn to the two circles then:
(1) T9 = TR
(3) TP_TR
(2) TPP = 4TR
(4) TP <TR
10
474 70. A, B, C, D are four points on a
circle. AC and BD intersect at a
point E such that ZBEC = 130°
and ZECD = 20°. ZBAC is
(1) 120° (2) 90°
(3) 100° (4) 110°
9
475 A circle of radius 7 is tangent to the
lines of an angle ( 60^{circ} . ) is larger circle of
radius ( r ) is tangent to same lines as
well as given circle, then value of ( r ) is:
begin{tabular}{l}
A ( .7 sqrt{3} ) \
hline
end{tabular}
B. ( frac{28}{sqrt{3}} )
( c cdot 21 )
D. 14
10
476 ( boldsymbol{O} ) is the centre of the circle having
radius ( 5 mathrm{cm} . A B ) and ( A C ) are two chords
such that ( A B=A C=6 mathrm{cm} . ) If ( 0 mathrm{A} )
meets ( B C ) at ( M, ) then ( O M ) is equal to
A . ( 3.6 mathrm{cm} )
B. ( 1.4 mathrm{cm} )
( c cdot 2 c m )
( 0.3 mathrm{cm} )
9
477 71. In the figure XAY is a tangent to
the circle with centre O at A. If
ZBAX=70°, ZBAQ = 40° then
ZABO is equal to :
UL
(1) 20°
(3) 35°
(2) 30°
(4) 40°
9
478 Find the centre and radius of the circle
( x^{2}+y^{2}-4 x-8 y-45=0 )
( mathbf{A} cdot(2,6), sqrt{63} )
B . ( (2,4), sqrt{65} )
c. ( (2,-4), sqrt{66} )
D. None
9
479 The radius of the circle is ( 25 mathrm{cm} ) and the length of one of its chord is ( 40 mathrm{cm} ). find
the distance of the chord from the
centre
9
480 If the line ( 3 x-4 y-8=0 ) divides the
circumference of the circle with centre
(2,-3) in the ratio ( 1: 2 . ) Then, the radius of the circle is
A.
B. 2
( c cdot 3 )
D. 4
9
481 The tangent to the circle ( x^{2}+y^{2}=9 )
which is parallel to y-axis and does not lie in third quadrant, touches the circle
at the point
A. (-3,0)
B. (3,0)
D. (0,-3)
10
482 f’o’ is the centre of the circle ; ( 0 L=4 )
( mathrm{cm}, mathrm{AB}=6 mathrm{cm} ) and ( mathrm{OM}=3 mathrm{cm}, ) then ( mathrm{CD} )
( A cdot 4 mathrm{cm} )
B. ( 8 mathrm{cm} )
( c cdot 6 mathrm{cm} )
D. ( 10 mathrm{cm} )
9
483 ( A B ) and ( C D ) are two equal chords of a
circle with centre ( boldsymbol{O} ) which intersect
each other at right angle at point ( P . ) If ( O M perp A B ) and ( O N perp C D ; ) show that
OMPN is a square
9
484 The length of chord of radius ( 25 mathrm{cm} ) and
Distance at ( 7 mathrm{cm} ) is
9
485 In the diagram 0 is the centre of a
circle. ( A E+E B=C E+E D O P perp A B ) and ( 0 Q )
( perp ) CD then true relation between OP and
OQ is
A. ор > ( 0 Q )
B. op < ( 0 Q )
( c cdot o p=frac{1}{2} o Q )
D. OP = OO
9
486 The figure is a circle with center ( O ) and
diameter ( 10 mathrm{cm}, P Q=1 mathrm{cm} . ) Find the
length of ( boldsymbol{R} boldsymbol{S} )
( mathbf{A} cdot 6 mathrm{cm} )
B. ( 4 mathrm{cm} )
( mathbf{c} .5 mathrm{cm} )
D. ( 3 mathrm{cm} )
9
487 If the tangent ( P Q ) and ( P R ) are drawn to
the circle ( x^{2}+y^{2}=a^{2} ) from the point
( Pleft(x_{1}, y_{1}right), ) then the equation of the
circumcircle of ( triangle boldsymbol{P Q R} ) is
A ( cdot x^{2}+y^{2}-x x_{1}-y y_{1}=0 )
B . ( x^{2}+y^{2}+x x_{1}+y y_{1}=0 )
c. ( x^{2}+y^{2}-2 x x_{1}-2 y y_{1}=0 )
D. None of these
9
488 Tangents drawn from the point ( boldsymbol{P}(mathbf{1}, boldsymbol{8}) )
to the circle ( x^{2}+y^{2}-6 x-4 y-11= )
0 touch the circle at the point ( A ) and ( B )
The equation of the circumcentre of the
( triangle boldsymbol{P} boldsymbol{A} boldsymbol{B} ) is
A ( cdot x^{2}+y^{2}+4 x-6 y+19=0 )
B . ( x^{2}+y^{2}-4 x-10 y+19=0 )
c. ( x^{2}+y^{2}-2 x+6 y-29=0 )
D. ( x^{2}+y^{2}+6 x-4 y+19=0 )
10
489 The number of common tangents to the
( operatorname{circles} x^{2}+y^{2}=4 ) and ( x^{2}+y^{2}-4 x+ )
( 2 y-4=0 ) is
A . 1
B . 2
( c .3 )
D. 4
10
490 The circle whose radius is ( 1 mathrm{cm} ) then the
diameter of the circle is
9
491 55. In below figure O is centre of
circle and ZAOB = 110° and
ZAOC = 90°. then ZBAC will
be equal to
1890
(1) 60
(3) 80
(2) 70
(4) 90°
9
492 Fill in the blanks with correct word(s) to
make the statement true:
A radius of a circle is a line segment with one end point at and the
other end point on
9
493 f an isosceles ( triangle A B C ) in which ( A B= )
( A C=6 mathrm{cm} ) is inscribed in a circle of
radius ( 9 mathrm{cm} . ) Find area of the triangle.
( mathbf{A} cdot 8 c m^{2} )
B. ( 8 sqrt{2} c m^{2} )
( c cdot 6 c m^{2} )
D. none
9
494 MN and MQ are tangents from a point ( mathrm{M} )
outside the given circle with center ( boldsymbol{O} )
If ( angle N O Q=120^{circ} ) then which of the
following rlations holds true:
A ( . N Q=M N=M Q )
в. ( N Q=O M )
c. ( O Q=O M )
D. ( O N=M N )
10
495 It two equal chords of a circle intersect within the circle. Prove that the line
joining the point of intersection to the centre makes equal angles with the chords.
9
496 If
( A B ) is a chord of a circle with centre ( O )
and ( P ) is a point on ( A B ) such that ( B P=4 P A, O P=5 mathrm{cm} ) and the radius
of the circle is ( 7 mathrm{cm} ), find the value of ( (sqrt{6} times A B) )
9
497 A chord of a circle of radius ( 12 mathrm{cm} )
subtends an angle of ( 120^{circ} ) at the center
Find the area of the corresponding segment of the circle.
9
498 67. The distance between two paral-
lel chords of length 8 cm each in
a circle of diameter 10 cm is
(1) 6 cm (2) 7 cm
(3) 8 cm (4) 5.5 cm
9
499 an infinite number of tangents can be
drawn from (1,2) to the circle ( x^{2}+y^{2}- )
( 2 x-4 y+lambda=0 ) then ( l a m b d a ) is
A . -20
B. 0
c. 5
D. can no be determined
10
500 A circle of radius ( 3 mathrm{cm} ) can be drawn
through two points ( A, B ) such that
( A B=6 mathrm{cm} )
State True or False
( A ). False
B. True
c. Cannot be determined
D. None of the above
9
501 ( boldsymbol{X Y}=? )
4.26
3.12
( c cdot 14 )
( D )
9
502 If a line intersects a circle in two
distinct points then it is known as a
A. chord
B. secant
c. tangent
D. segment
10
503 57. AB and CD (AB||CD) are the two
chord of a circle with length 5
cm and 11 cm respectively. If the
distance between AB and CD is
3 cm, then the radius of circle
will be
(1) 1104 cm (2) 194 cm
cm
cm
9
504 f the line ( x cos alpha+y sin alpha=p )
represents the common chord ( A P Q B ) of
the circles ( x^{2}+y^{2}=a^{2} ) and ( x^{2}+y^{2}= )
( b^{2}(a>b) ) as shown in the figure, then
( A P ) is equal to
A ( cdot sqrt{a^{2}+p^{2}}+sqrt{b^{2}+p^{2}} )
B. ( sqrt{a^{2}-p^{2}}+sqrt{b^{2}-p^{2}}^{2} )
c. ( sqrt{a^{2}-p^{2}}-sqrt{b^{2}-p^{2}} )
D. ( sqrt{a^{2}+p^{2}}-sqrt{b^{2}+p^{2}}^{2} )
9
505 A regular hexagon & a regular dodecagon are inscribed in the same circle. If the side of the dodecagon is ( (sqrt{3}-1), ) then the side of the hexagon
is
A ( cdot sqrt{2}+1 )
B. ( frac{sqrt{3}+1}{2} )
c. 2
D. ( sqrt{2} )
9
506 Which of the following is secant to the circle given above?
( A cdot A B )
B. CD
( c cdot c )
( D . P O )
10
507 The line segment joining any two points on a circle is called a or an
A. arc of the chord
B. radius of the circle
c. chord of the circle
D. tangent of the circle
9
508 The radius of a circle is ( 17.0 mathrm{cm} ) and the
length of perpendicular drawn from its centre to a chord is ( 8.0 mathrm{cm} . ) Calculate
the length of the chord.
9
509 Explain the followings:
Chord
9
510 Chord ( A B ) of the circle ( x^{2}+y^{2}=100 )
passes through the point (7,1) and
subtends an angle of ( 60^{circ} ) at the
circumference of the circle. If ( m_{1} ) and
( m_{2} ) are the slopes of two such chords
then the value of ( m_{1} m_{2}, ) is
A . -1
B. 1
c. ( frac{7}{12} )
D. – 3
9
511 Line ( 3 x-4 y=k ) will cut the circle
( x^{2}+y^{2}-2 x+4 y-11=0 ) at distinct
points, if
A ( cdot k>frac{25}{7} )
в. ( 15<k<30 )
c. ( -9<k<31 )
D. None of these
10
512 In a circle whose radius is 10 inches, a
chord is 6 inches from the center. What
is the length of the chord?
A. 4 inches
B. 6 inches
c. 8 inches
D. 16 inches
9

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