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.

Circles Questions

List of circles Questions

Question NoQuestionsClass
1Find the values of ( x ) and ( y ) in the figures
given below
9
2The 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
3If 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
4n 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
5The 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
6What 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
7Circle 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
862. 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
9The 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
10The 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
11The 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
12How 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
1372. 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
14Find 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
15If ( 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
1668. 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
17If 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
18Chords ( 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
19Two 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
20A perpendicular at the end of the radius of a circle is
A. diameter
B. tangent
c. chord
D. anyline
10
21The tangent drawn at the end point of two pependicular diameter of a circle. prove that ( mathrm{PQ} ) and ( mathrm{RS} ) are parallel10
22A 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
23In 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
24A 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
25n 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
26Angle inscribed in a semi-circle is
( mathbf{A} cdot pi / 2 )
в. ( pi / 3 )
c. ( pi / 4 )
D.
9
27Determine the maximum number of
common tangents that can be drawn for each pair of circles shown.
10
28Circle 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
29The 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
30Length 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
31In 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
32In 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
33Suppose 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
3569. 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
36If ( 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
37If 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
38The 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
39Two 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
40If 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
41Find the measure of arc ( C D ).
A ( cdot 105 )
B . ( 55^{circ} )
( c cdot 108 )
D. 75
9
42Find 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
43The 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
44Length 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
45Which 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
46n 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
47In 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
48A 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
49If 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
50Draw a circle and mark a point in its
interior.
9
51f ( 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
52Draw a circle and mark a radius.9
53Determine 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
5474. ‘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
55In 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
56Find 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
57Find 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
58Statement:-Tangent at any point of a circle is perpendicular to the radius through the point of contact
If yes enter ( 1, ) else 0
10
59Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.9
60Perimeter of a circle is called as:
A . area
B. circumference
c. volume
D. none
9
61The 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
63Fill in the blanks:
The diameter of a circle are
9
64In the given figure, ( O ) is the centre of
circle, ( angle A E C=40^{circ}, ) then find the value
of ( a+b+c )
9
65In 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
66If ( 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
67The 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
68Find 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
69The 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
70The 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
71Find the value of ( x )
( A, x=6 )
B . ( x=7 )
c. ( x=8 )
( x=9 )
9
72If 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
73If 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
74If 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
75Find the centre and radius of each of the
following circle. ( (x-5)^{2}+(y-3)^{2}=20 )
9
76Three 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
77Which 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
78A 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
79Prove that the centre of a circle
touching two intersecting lines lies on the angle bisector of the lines.
10
80What 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
81Tangents 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
82Define the following term of the circle. Chord9
83In 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
84Two 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
85Find the value of ( q )
( A cdot 3 sqrt{5} )
B. ( 3 sqrt{10} )
( c cdot 2 sqrt{10} )
D. ( 8 sqrt{10} )
10
86The 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
87Given, 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
88Find 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
89What is chord?9
90A line that intersects a circle at two
distinct points is called
A . a diameter
B. a secant
c. a tangent
D. a radius
10
91The 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
92A 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
93The 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
94The perpendicular from the centre of a circle to a chord bisects the chord.9
95Find 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
96Draw a pair of tangents to a circle of
radius 5 cm which are inclined to each
other at an angle of 60
10
97In 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
98Equal chords of a circle subtend equal
angle on centre
A . True
B. False
9
99Define congruent chords.9
100If 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
101Draw 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
103The 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
104If 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
105In 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
106In 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
107Find the diameter of the circle if its.
Circumference is ( 62.8 mathrm{cm}(pi=3.14) )
9
108n 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
109The 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
110The 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
111In 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
112A pair of opposite sides of a cyclic quadrilateral are equal. Prove that its diagonal are also equal9
113Two 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
114Tangents ( 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
115CP 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
116Through 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
117In 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
118If 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
119statement-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
120The 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
121In 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
122n 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
123Find the centre and radius of the circle
( x^{2}+y^{2}-4 x-8 y-45=0 )
9
124In 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
125Recall 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
126Each 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
128A 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
12969. 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
130Line 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
131In 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
132Equation 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
133If 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
134what is tangent of a circle and
definition?
10
135If 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
136A 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
137The tangents drawn at the ends of a diameter of a circle are ?
A. perpendicular
B. parallel
c. adjacent
D. none of the above
10
138In 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
139In 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
140In the figure, 0 is the centre of the circle
Find the length of ( mathrm{CD} ), if ( mathrm{AB}=5 mathrm{cm} )
9
14169. 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
142Assertion
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
143The 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
144For 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
145In 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
146If ( 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
147State 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
148From 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
149Show 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
150The 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
151In 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
152Define diameter.9
153Prove 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
15468. 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
155In 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
156A 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
157Chords 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
158Find 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
159Find 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
160If ( 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
161Prove that if chords of congruent circles subtend equal angle at their centres, then the chords are equal.9
162Find the value of ( x+y ) in the given
figure (in degrees)
10
163From the figure, identify a sector9
164Recall 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
165In 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
166Draw any circle and mark an arc.9
167Find 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
168If 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
169In 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
170A secant intersects the circle at
points.
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
( D )
10
171Tangents 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
17266. 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
173The 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
174Find the equation of the tangent to the
curve ( y=x^{2}-7 ) at the point where it
cuts the ( y ) – axis.
10
175Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at a
center.
10
176The 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
178In 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
179A 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
180Find the center and radius of the circle.
( (x+5)^{2}+(y-3)^{2}=36 )
9
181What 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
182The 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
184circle 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
18562. 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
186A 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
187In 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
188The 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
189A 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
190Write 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
191Illustration 2.21 Find the length of an arc of a circle of
radius 5 cm subtending a central angle measuring 15º.
9
192n 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
193Two 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
194n 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
195STATEMENT – 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
196A 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
198Find 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
199The 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
200Three 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
201Find the centre and radius of the circle
( 2 x^{2}+2 y^{2}=3 x-5 y+7 )
9
202Out 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
203The 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
204What 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
205If 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
20656. 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
207The 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
209If 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
21069. 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
211Prove that the tangents drawn from an
external point to a circle are equal.
10
212In 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
213The 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
214The longest chord passes through a centre of a circle is9
215In 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
21675. 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
217In 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
218Suppose you are given a circle. Give steps of construction to find its centre.9
219n 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
220In 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
221ACB 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
222Find the value of ( c ) if (2,3) lies on the circle ( x^{2}+y^{2}+2 x+3 y+c=0 )9
223Prove: If a chord of circle ( x^{2}+y^{2}=8 )
makes equal intercepts of length ‘a’ on the coordinate axes then ( |a|<4 )
9
224If ( 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
225The 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
226The 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
227O’ 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
228Find 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
229Prove that the parallelogram
circumscribing a circle is a rhombus.
10
230If 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
231In 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
23273. PO is a tangent to the circle at R
then mZPRS is equal to :
BOT
(1) 30°
(3) 60°
(2) 40°
(4) 80°
9
233The 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
234In 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
235Chord ( 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
236A 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
237Find the radius of the circle passing the points (0,0),(1,0) and (0,1)9
238Tangent ( 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
239The 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
24059. 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
241If ( 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
24271. 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
243Circumference 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
244f 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
245Find diameter ( & ) circumference with
radius ( 7.7 mathrm{cm} )
9
246In 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
247In 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
248A 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
249The 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
251Find the total cost of wooden fencing around a circular garden of diameter ( 28 m . ) If ( 1 m ) of fencing costs 23009
252The 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
253Let ( 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
255In 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
25667. 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
257The 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
258In 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
259Consider 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
26070. 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
261What 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
262The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord?9
263Prove 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
26458. 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
265The 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
266Recall 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
267A 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
268Line segment joining the centre to any point on the circle is a radius of the circle.
A. True
B. False
9
269Given 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
270From 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
271Find ( Q M )
( A cdot 13 c m )
3. ( 12 mathrm{cm} )
( c .5 mathrm{cm} )
D. ( 8 mathrm{cm} )
9
272Find 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
273Consider 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
274The 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
275If ( 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
276If 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
277Circles 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
278A 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
27969. 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
280The 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
281A 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
282f 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
283Two 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
284A 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
285If 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
286Prove that the line joining the mid-
points of two parallel chords of a circle
passes through the centre.
9
287Draw two tangents from a point ( 5 mathrm{cm} ) away from the centre of a circle of
radius ( 3 mathrm{cm} )
10
288Find the value of ( x ) in each of the
following diagrams
( (mathbf{i}) )
(ii)
9
289If 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
290Equation 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
291If 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
292An equilateral triangle is inscribed in a circle of radius ( 6 mathrm{cm} . ) Find its side.9
293The 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
294A 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
29565. 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
296A 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
297From the figure, identify a point in the
exterior.
9
298n 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
299In the given figure, find the value of ( x )
( A cdot 25 )
3.30
( c .35 )
2.4
10
300Two 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
301Fill in the blanks
The longest chord of a circle is a
of the circle.
9
302Area 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
303Let ( 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
304Two 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
306In 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
307If 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
308In 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
311The 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
312Given 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
313Two 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
314A 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
315Prove that if chords of congruent circles
subtend equal angles their centres, then the chords are equal.
9
316Find 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
31869. 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
320The 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
32159. 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
322If 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
323In 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
324Prove that the line joining a mid point of
a chord to the centre of circle is
perpendicular to it
9
325Hilswer 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
326Find 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
327Prove that the lengths of the tangents drawn from an external point to a circle
are equal.
10
32869. 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
329A 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
330Draw a circle and mark a diameter.9
331The 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
333In 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
33470. 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
335Find the value of ( x )
A ( cdot 50^{circ} )
B ( .60^{circ} )
( c cdot 70^{circ} )
D. ( 80^{circ} )
10
336Find the angle marked ( a )
4.77
8. 36
( c cdot 41^{circ} )
( mathbf{D} cdot 13^{circ} )
10
337The 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
338In 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
339n 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
340Tangents 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
341Name the following part from the adjacent figure where ‘O’ is the center of
the circle.
( boldsymbol{A O} )
9
342n 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
343Prove that if chords of congruent circles subtend equal angles at their centre, then the chords are equal.9
344In two concentric circle, prove that all
chords of the outer circle which touch
the inner are of equal length.
9
345Find 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
346Prove that the angle in a semicircle is a
right angle.
9
34767. 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
348If ( 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
349In 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
350Two 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
351A 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
352In a diagram ( boldsymbol{O} ) is the centre of circle.
Calculate the value ( a )
A . 43
B. 53
( c cdot 63 )
D. 33
9
353In 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
354n 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
355Establish the formula for area and
circumference of circle.
9
356If ( 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
357A 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
35855. 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
359If 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
361The 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
362A 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
363Draw 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
364Coordinates 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
365Two 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
366What is a line passing through two points on a circle called?
A. secant
B. Digonal
c. Radius
D. tangent
10
367The 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
368In 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
369Tangents 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
370If 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
37165. 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
372n 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
373OA.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
374In 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
375f 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
376The 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
377Find 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
378Consider 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
379Perimeter of a circle is called its
A . circumference
B. area
c. diameter
D. none of these
9
380n 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
382Calculate 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
383In 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
384Distance 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
38552. 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
386If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of other chord9
38771. 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
388The 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
389The 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
390In 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
391Draw any circle and mark a sector9
392Prove that the line joining the mid-
points of two equal chords of a circle subsent equal angles with the chord.
9
39371.
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
394Tangents 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
395In 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
396Find the centre and radius of the circle
( x^{2}+y^{2}+6 x+8 y-96=0 )
9
397A 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
398If two circles intersect at two points, prove that their centres lie on the
perpendicular bisector of the common chord
9
399Prove 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
400In 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
401If 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
402If 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
404In 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
405Prove 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
406The
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
40867. 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
409Two 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
410Two 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
411Which is a secant?
( A cdot ) ми
B. on
( c . P Q )
D. None
10
41270. 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
413Find the area of the sector of a circle
whose radius is ( 14 mathrm{cm} ) and angle of
sector is ( 45^{circ} )
9
414The 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
415Find 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
416If ( 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
417What 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
418If 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
419In 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
42070. 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
421Find ( 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
423The 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
424Given ( 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
425The common point of the tangent and circle is called
A. Intersecting points
B. Secant
c. Point of contact
D. None
10
426The 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
427n the figure, ( P Q=R S ) and ( angle O R S= )
( 48^{circ} ) Find ( angle O P Q ) and ( angle R O S )
9
428The 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
429In 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
430From 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
43166. 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
432Find the length of longest chord of the circle if radius is ( 2.9 mathrm{cm} ) in ( mathrm{cm} )9
433At 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
434Two 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
435In 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
437The 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
438In 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
439The 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
440The 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
441In 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
442Length 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
443A 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
444If 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
445The 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
446If ( 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
4478.
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
448In which circles, angles at the centers make a equal chords?
A. concentric circles
B. eccentric circles
c. tangential circles
D. equal circles
9
449A 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
450Three 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
451The distance, once around the circle is
called
A. diameter
B. center
c. circumference
D. chord
9
452Two 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
454The 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
456In the figure, if ( boldsymbol{A B}=boldsymbol{C D} ) and
( angle A O B=90^{circ} ) find ( angle C O D )
9
457In 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
458Find the length of chord of circle with
radius ( 5 c m ) and distance from center
( 2 c m )
9
459Find the value of ( x )
( mathbf{A} cdot x=10 )
B. ( x=8 )
( mathbf{c} cdot x=6 )
D. ( x=3 )
9
460i) A circle can have ( _{–} ) -parallel tangents.
ii) The point common to the tangent and the circle is called
10
461The 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
462n 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
463Two 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
46423. 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
465Find 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
466The 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
468If ( 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
469If the diameter of circle is ( 10 mathrm{cm}, ) then find the radius of circle.9
470The 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
471In 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
472Find the radius of that circle whose area
is ( 616 mathrm{cm}^{2}(text { in } mathrm{cm} .) )
9
47367. 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
47470. 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
475A 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
47771. 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
478Find 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
479The 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
480If 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
481The 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
482f’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
484The length of chord of radius ( 25 mathrm{cm} ) and
Distance at ( 7 mathrm{cm} ) is
9
485In 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
486The 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
487If 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
488Tangents 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
489The 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
490The circle whose radius is ( 1 mathrm{cm} ) then the
diameter of the circle is
9
49155. 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
492Fill 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
493f 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
494MN 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
495It 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
496If
( 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
497A 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
49867. 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
499an 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
500A 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
502If a line intersects a circle in two
distinct points then it is known as a
A. chord
B. secant
c. tangent
D. segment
10
50357. 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
504f 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
505A 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
506Which of the following is secant to the circle given above?
( A cdot A B )
B. CD
( c cdot c )
( D . P O )
10
507The 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
508The 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
509Explain the followings:
Chord
9
510Chord ( 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
511Line ( 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
512In 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

Hope you will like above questions on circles and follow us on social network to get more knowledge with us. If you have any question or answer on above circles questions, comments us in comment box.

Stay in touch. Ask Questions.
Lean on us for help, strategies and expertise.

Leave a Reply

Your email address will not be published. Required fields are marked *