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

#### List of circles Questions

Question No | Questions | Class |
---|---|---|

1 | Find the values of ( x ) and ( y ) in the figures given below | 9 |

2 | The range of values of ( a ) such that the angle ( theta ) between the pair of tangents drawn from ( (a, 0) ) to the circle ( x^{2}+ ) ( y^{2}=1 ) satisfies ( frac{pi}{2}<theta<pi ), is A ( .(1,2) ) B. ( (1, sqrt{2}) ) c. ( (-sqrt{2},-1) ) (年) ( (-sqrt{2},-1) ) D. ( (-sqrt{2},-1) cup(1, sqrt{2}) ) | 10 |

3 | If the length of the common chord of two intersecting equal circles be ( 6 mathrm{cm} ) and if the radius of each circle be ( 5 mathrm{cm} ) then the distance between the centers of the circle is A. ( 7 mathrm{cm} ) B. ( 8 mathrm{cm} ) ( c .9 mathrm{cm} ) D. none | 9 |

4 | n given figure ( angle P Q R=100^{circ}, ) where ( P, Q ) and ( R ) are points on a circle with centre ( O . ) Then, ( angle O P R ) is ( A cdot 20 ) B. 10 ( c cdot 30^{circ} ) D. 40 | 9 |

5 | The coordinate that the chord ( x cos alpha+ ) ( boldsymbol{y} sin boldsymbol{alpha}-boldsymbol{p}=mathbf{0} ) of ( boldsymbol{x}^{2}+boldsymbol{y}^{2}-boldsymbol{a}^{2}=mathbf{0} ) may subtend a right angle at the centre of the circle is? A ( cdot a^{2}=2 p^{2} ) B ( cdot p^{2}=2 a^{2} ) c. ( a=2 p ) D. ( p=2 a ) | 9 |

6 | What will be ( angle X O Y ) if arc ( A B= ) arc ( X Y ) and ( angle A O B=60^{circ} ? ) ( A cdot 30^{circ} ) B. ( 45^{circ} ) ( c cdot 50^{circ} ) D. 60 | 9 |

7 | Circle O has diameters AB and CD perpendicular to each other. AM is any chord intersecting ( mathrm{CD} ) at ( mathrm{P.} ) Then ( A P . overline{A M} ) is equal to: A. ( overline{A O} . overline{O B} ) в. ( overline{A O} . overline{A B} ) ( c cdot overline{C P} cdot overline{C D} ) D. ( overline{C P} . overline{P D} ) ह. ( overline{c O} . overline{O P} ) | 9 |

8 | 62. ABCD is a cyclic trapezium such that AD||BC, if ZABC = 70°, then the value of ZBCD is : (1) 60 (2) 70° (3) 40 (4) 80° | 9 |

9 | The slope of the tangent to the curve ( y=int_{0}^{x} frac{d t}{1+t^{3}} ) at the point where ( x=1 ) is. A ( cdot frac{1}{4} ) B. ( frac{1}{3} ) ( c cdot frac{1}{2} ) D. 1 | 10 |

10 | The radius of a circle is given as ( 15 mathrm{cm} ) and chord AB subtends an angle of ( 131^{circ} ) at the centre ( C ) of the circle.Using trigonometry ,calculate: (i) the length of ( A B ) (ii) the distance of ( A B ) from the centre ( C ) | 9 |

11 | The locus of the centre of the circles which touch both the circles ( x^{2}+y^{2}= ) ( a^{2} ) and ( x^{2}+y^{2}=4 a x ) externally has the equation: A ( cdot 12(x-a)^{2}-4 y^{2}=3 a^{2} ) B ( cdot 9(x-a)^{2}-5 y^{2}=2 a^{2} ) C ( cdot 8 x^{2}-3(y-a)^{2}=9 a^{2} ) D. None of these | 9 |

12 | How many tangents can be drawn on the circle of radius ( 5 mathrm{cm} ) form a point lying outside the circle at distance ( 9 mathrm{cm} ) from the center | 10 |

13 | 72. Two circles are of radii 7 cm and 2 cm their centres being 13cm apart. Then the length of direct common tangent to the circles between the points of contact is (1) 12 cm (2) 15 cm (3) 10 cm (4) 5 cm | 10 |

14 | Find the values of ( x ) and ( y ) A . ( x=10.3, y=12.7 ) B . ( x=12.9, y=15.6 ) C. ( x=15.3, y=12.3 ) D. ( x=19.3, y=15.4 ) | 9 |

15 | If ( 9.2 mathrm{cm} ) is the diameter of a circle then its radius is A ( .4 .1 mathrm{cm} ) в. ( 4.6 mathrm{cm} ) c. ( 4.8 mathrm{cm} ) D. ( 4.3 mathrm{cm} ) | 9 |

16 | 68. AB = 8 cm and CD = 6 cm are two parallel chords on the same side of the centre of a circle. The distance between them is 1 cm. The radius of the circle is (1) 5 cm (2) 4 cm (3) 3 cm (4) 2 cm | 9 |

17 | If the angle between two tangents drawn from an external point ( P ) to a circle of radius a and a center ( O ), is ( 60^{circ} ) then find the length of ( O P ) | 10 |

18 | Chords ( M N ) and ( R S ) of a circle intersect at ( boldsymbol{P} ) outside the circle If ( boldsymbol{P N}=mathbf{3} boldsymbol{c m}, boldsymbol{M} boldsymbol{N}=mathbf{5} boldsymbol{c m}, boldsymbol{P} boldsymbol{R}=boldsymbol{2} boldsymbol{c m} ) then the value of ( S R ) is equal to ( mathbf{A} cdot 5 mathrm{cm} ) B. ( 8 mathrm{cm} ) c. ( 15 mathrm{cm} ) D. ( 10 mathrm{cm} ) | 9 |

19 | Two parallel chords ( A B ) and ( C D ) are 3.9 cm apart and lie on opposite sides of the centre of a circle. If ( A B=1.4 mathrm{cm} ) and ( C D=4 mathrm{cm}, ) find the radius of the circle. A. ( 3 mathrm{cm} ) B. ( 3.2 mathrm{cm} ) ( c .2 .3 mathrm{cm} ) D. ( 2 mathrm{cm} ) | 9 |

20 | A perpendicular at the end of the radius of a circle is A. diameter B. tangent c. chord D. anyline | 10 |

21 | The tangent drawn at the end point of two pependicular diameter of a circle. prove that ( mathrm{PQ} ) and ( mathrm{RS} ) are parallel | 10 |

22 | A straight line ( x=y+2 ) touches the ( operatorname{circle} 4left(x^{2}+y^{2}right)=r^{2} . ) The value of ( r ) is A ( cdot sqrt{2} ) B. ( 2 sqrt{2} ) ( c cdot 2 ) ( D ) | 9 |

23 | In the given figure, if 0 is the centre of a circle, ( P Q ) is a chord and the tangent ( P R ) at ( P ) makes an angle of ( 50^{circ} ) with ( P Q ) find ( angle P O Q ) ( A cdot 40 ) в. 80 ( c cdot 100 ) D. 120 | 10 |

24 | A point ( A ) is ( 26 c m ) away from the centre of a circle and the length of tangent drawn from ( A ) to the circle is 24 cm. Find the radius of the circle. ( mathbf{A} cdot 10 mathrm{cm} ) B. ( 20 mathrm{cm} ) ( mathbf{c} cdot 25 c m ) D. ( 15 mathrm{cm} ) | 10 |

25 | n given figure triangle ( mathrm{ABCCCC} ) circumscribes the circle with center 0 and radius ( 2 mathrm{cm} ) Area of ( Delta A B C ) is ( 16 mathrm{cm}^{2} ). find ( mathrm{AB} ) ( 5 mathrm{cm} ) ( 6 c ) ( 7 mathrm{cm} ) | 10 |

26 | Angle inscribed in a semi-circle is ( mathbf{A} cdot pi / 2 ) в. ( pi / 3 ) c. ( pi / 4 ) D. | 9 |

27 | Determine the maximum number of common tangents that can be drawn for each pair of circles shown. | 10 |

28 | Circle with centre 0 and radius 25 cms has a chord ( A B ) of length of 14 cms in it. Find the area of triangle AOB? | 9 |

29 | The points of intersection of the line ( 4 x-3 y-10=0 ) and the circle ( x^{2}+y^{2}-2 x+4 y-20=0 ) are………………..and. This question has multiple correct options A ( .(4,2) ) в. (-2,-6) D. (-2,-4) | 10 |

30 | Length of the chord joining the points ( P(alpha) ) and ( Q(beta) ) on the circle ( x^{2}+y^{2}= ) ( a^{2} ) is A ( cdot cos left(frac{alpha-beta}{2}right) ) в. ( 2 a sin left(frac{alpha-beta}{2}right) ) c. ( 2 a tan left(frac{alpha-beta}{2}right) ) D. ( 2 a csc left(frac{alpha-beta}{2}right) ) | 9 |

31 | In a circle with centre ( 0, O D perp ) chord ( A B ) If BC is the diameter, then ( mathbf{A} cdot A C=B C ) В ( . O D=B D ) c. ( A C=2 O D ) D. none of these | 9 |

32 | In a circle of radius ( 13 mathrm{cm}, P Q ) and ( R S ) are two parallel chords of length ( 24 mathrm{cm} ) and IOcm respectively. The chords are on the opposite sides of the centre. The distance between the chords is? A. ( 7 mathrm{cm} ) B. ( 17 mathrm{cm} ) ( c cdot 26 c m ) D. ( 12 mathrm{cm} ) | 9 |

33 | Suppose you are given a circle. Give a construction to find its centre. | 9 |

34 | ( O ) is the centre of the circle having radius ( 5 mathrm{cm} . O M perp ) chord ( A B . ) If ( boldsymbol{O} boldsymbol{M}=mathbf{4} mathrm{cm}, ) then the length of the chord ( A B ) is A. ( 6 mathrm{cm} ) B. ( 5 mathrm{cm} ) ( c cdot 8 mathrm{cm} ) D. ( 10 mathrm{cm} ) | 9 |

35 | 69. The distance between the cen- tres of the two circles with radii 4 cm and 9 cm is 13 cm. The length of the direct common tan- gent (between two points of con- tact) is (1) 13 cm (2) 153 cm (3) 12 cm (4) 18 cm | 9 |

36 | If ( P ) is a point on a circle with centre ( C ) then the line drawn through ( P ) and perpendicular to CP is the tangent to the circle at the point ( P ) A. True B. False c. Either D. Neither | 10 |

37 | If the diameter of a circle decreases to its ( frac{1}{4} ) then its radius decreases to A ( cdot frac{1}{2} ) B. 4 ( c cdot frac{1}{4} ) D. | 9 |

38 | The radius of a circle with centre 0 is 7 ( mathrm{cm} . ) Two radii OA and ( mathrm{OB} ) are drwan at right angles to each other. Find the areas of minor and major segments. | 9 |

39 | Two chords ( A B ) and ( C D ) of lengths ( 5 mathrm{cm} ) and ( 11 mathrm{cm} ) respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between ( A B ) and ( C D ) is ( 6 mathrm{cm} ) find the radius of the circle. | 9 |

40 | If two parallel chords on the same side of the centre of a circle are ( 6 mathrm{cm} ) and 8 ( mathrm{cm}, ) and they are ( 1 mathrm{cm} ) apart, then the diameter of the circle will be ( mathbf{A} cdot 14 mathrm{cm} ) B. ( 10 mathrm{cm} ) c. ( 8 mathrm{cm} ) D. ( 5 mathrm{cm} ) | 9 |

41 | Find the measure of arc ( C D ). A ( cdot 105 ) B . ( 55^{circ} ) ( c cdot 108 ) D. 75 | 9 |

42 | Find the points of intersection of the line ( x-y+2=0 ) and the circle ( 2 x^{2}+ ) ( 2 y^{2}-29 x-19 y+56=0 . ) Also determine the length of the chord intercepted. | 9 |

43 | The line ( 4 y-3 x+lambda=0 ) touches the circle ( x^{2}+y^{2}-4 x-8 y-5=0 . ) The value of ( lambda ) is ( mathbf{A} cdot 29 ) B . 10 c. -35 D. None of these | 10 |

44 | Length of a chord of a circle is ( 24 mathrm{cm} ) and its distance from the centre is 5 ( mathrm{cm} . ) Find the diameter of the circle. | 9 |

45 | Which of the following is/ are correct? This question has multiple correct options A. A line segment with its endpoints lying on a circle is called a chord of the circle. B. A line that intersects a circle at exactly one point is called a tangent to the circle. C. Angle in a semi-circle is a right angle. D. Lengths of the two tangents to a circle from an external point are equal | 10 |

46 | n figure, chords ( overline{P Q} ) and ( overline{R S} ) intersect at ( mathrm{T} . ) If ( boldsymbol{m} angle boldsymbol{R}=mathbf{5 0}^{boldsymbol{o}} ) and ( boldsymbol{m} angle boldsymbol{P}=mathbf{4 6}^{boldsymbol{o}} ) the number of degrees in minor arc PR is ( A cdot 84 ) B. 168 ( c cdot 42 ) D. 130 E. cannot be determine | 9 |

47 | In a circle whose radius is ( 8 mathrm{cm}, ) a chord is drawn at a point ( 3 mathrm{cm} ). from the centre of the circle. The chord is divided into two segments by a point on it. If one segment of the chord is ( 9 mathrm{cm}, ) What is the length of the other segment? | 9 |

48 | A circle has two equal chords ( A B ) and ( A C, ) chord ( A D ) bisects ( B C ) in ( E . ) If ( A C=12 ) and ( A E=8 c m, ) then the measure of ( A D ) is ? A ( .24 mathrm{cm} ) B. ( 18.5 mathrm{cm} ) c. ( 18 mathrm{cm} ) D. ( 19 mathrm{cm} ) | 9 |

49 | If radii of two concentric circles are 4 ( mathrm{cm} ) and ( 5 mathrm{cm}, ) then the length of each chord of one circle which is tangent to the circle is A. ( 3 mathrm{cm} ) B. ( 6 mathrm{cm} ) ( c .9 mathrm{cm} ) D. ( 1 mathrm{cm} ) | 9 |

50 | Draw a circle and mark a point in its interior. | 9 |

51 | f ( boldsymbol{m}(boldsymbol{a} boldsymbol{r} boldsymbol{c} boldsymbol{B} boldsymbol{C})=boldsymbol{8} boldsymbol{0}^{o}, ) find ( boldsymbol{m}(boldsymbol{a} boldsymbol{r} boldsymbol{c} boldsymbol{C} boldsymbol{D}) ) A ( cdot 40^{circ} ) B. ( 80^{circ} ) ( c cdot 120^{circ} ) D. ( 140^{circ} ) | 9 |

52 | Draw a circle and mark a radius. | 9 |

53 | Determine the length of the chord common to the circles ( x^{2}+y^{2}= ) ( 64 a n d x^{2}+y^{2}-16 x=0 ) A ( cdot 2 sqrt{3} ) B. ( 4 sqrt{3} ) ( c cdot 6 sqrt{3} ) D. ( 8 sqrt{3} ) | 9 |

54 | 74. ‘O’ is the circumcentre of trian- gle ABC. If Z BAC = 50° then Z OBC is (1) 50° (2) 100 (3) 130° (4) 40° | 9 |

55 | In the diagram, ( O ) is the centre of the circle. Given that ( O Q= ) ( 5 mathrm{cm} ) and ( A N=8 mathrm{cm}, ) find the length of ( boldsymbol{P Q} ) | 9 |

56 | Find the length of a chord that is at a distance of ( 5 mathrm{cm} ) form the centre of a circle of radius ( 13 mathrm{cm} ) ( A cdot 20 mathrm{cm} ) B . ( 24 mathrm{cm} ) c. ( 15 mathrm{cm} ) D. ( 12 mathrm{cm} ) | 9 |

57 | Find the equation of tangent at (3,4) for the circle ( x^{2}+y^{2}=25 ) A. ( 3 x+4 y=25 ) в. ( 3 x-4 y=25 ) c. ( 3 x+4 y+25=0 ) D. ( 4 x-3 y=25 ) | 10 |

58 | Statement:-Tangent at any point of a circle is perpendicular to the radius through the point of contact If yes enter ( 1, ) else 0 | 10 |

59 | Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal. | 9 |

60 | Perimeter of a circle is called as: A . area B. circumference c. volume D. none | 9 |

61 | The length of the common chord of two intersecting circles is ( 30 mathrm{cm} ). If the radii of the two circles are ( 25 mathrm{cm} ) and ( 17 mathrm{cm} ) find the distance (in cm) between their centres. | 9 |

62 | ( O ) is the center of the circle. ( O P=12 ) ( mathrm{cm}, ) and ( boldsymbol{O B}=mathbf{1 3} mathrm{cm} . ) Find ( boldsymbol{A B} ) A. ( 8 mathrm{cm} ) B. ( 10 mathrm{cm} ) ( mathrm{c} cdot 12 mathrm{cm} ) D. ( 13 mathrm{cm} ) | 9 |

63 | Fill in the blanks: The diameter of a circle are | 9 |

64 | In the given figure, ( O ) is the centre of circle, ( angle A E C=40^{circ}, ) then find the value of ( a+b+c ) | 9 |

65 | In the diagram ( O ) is the centre of the circle with diameter ( 20 mathrm{cm} ) The circle is the locus of a point ( boldsymbol{X} ) State the distance of ( X ) from ( O ) ( A cdot 5 mathrm{cm} ) B. ( 8 mathrm{cm} ) ( c .10 mathrm{cm} ) D. ( 20 mathrm{cm} ) | 9 |

66 | If ( 9.2 mathrm{cm} ) is the diameter of the circle, then its radius is A ( .4 .1 mathrm{cm} ) B. ( 4.6 mathrm{cm} ) c. ( 4.8 mathrm{cm} ) D. ( 4.3 mathrm{cm} ) | 9 |

67 | The locus of the mid points of the chord of the circle ( x^{2}+y^{2}=4, ) which subtended a right angle at the origin is A. ( x+y=1 ) B . ( x^{2}+y^{2}=1 ) c. ( x+y=2 ) D. ( x^{2}+y^{2}=2 ) | 9 |

68 | Find the distance of a perpendicular from the centre of a circle to the chord if the diameter of the circle is ( 30 mathrm{cm} ) and its chord is ( 24 mathrm{cm} ) ( mathbf{A} cdot 6 mathrm{cm} ) в. ( 7 mathrm{cm} ) ( mathrm{c} .9 mathrm{cm} ) D. ( 10 mathrm{cm} ) | 9 |

69 | The length of the chord ( x+y=3 ) intercepted by the circle ( x^{2}+y^{2}- ) ( 2 x-2 y-2=0 ) is A ( cdot frac{7}{2} ) B. ( frac{3 sqrt{3}}{2} ) c. ( sqrt{14} ) D. ( frac{sqrt{7}}{2} ) | 9 |

70 | The radius of any circle touching the ( operatorname{lines} 3 x-4 y+5=0 ) and ( 6 x-8 y- ) ( mathbf{9}=mathbf{0} ) is A ( cdot frac{19}{10} ) в. ( frac{19}{20} ) c. ( frac{9}{20} ) D. ( frac{90}{20} ) | 10 |

71 | Find the value of ( x ) ( A, x=6 ) B . ( x=7 ) c. ( x=8 ) ( x=9 ) | 9 |

72 | If two parallel chords of length ( 8 mathrm{cm} ) and ( 6 mathrm{cm} ) in a circle of radius ( 5 mathrm{cm} ) are on the opposite sides of the center then the distance between the parallel chords is A . ( 5 mathrm{cm} ) в. 6 ст ( c .7 mathrm{cm} ) D. None of these | 9 |

73 | If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel. | 9 |

74 | If tangents ( boldsymbol{P} boldsymbol{A} ) and ( boldsymbol{P} boldsymbol{B} ) from a point ( boldsymbol{P} ) to a circle with centre ( O ) are inclined to each other at angle of ( 80^{circ}, ) then ( angle P O A ) is equal to. A ( .50^{circ} ) B. ( 60^{circ} ) ( c cdot 70^{circ} ) D. ( 80^{circ} ) | 10 |

75 | Find the centre and radius of each of the following circle. ( (x-5)^{2}+(y-3)^{2}=20 ) | 9 |

76 | Three schools situated at ( P, Q ) and ( R ) in the figure are equidistant from each other as shown in the figure. Find ( angle Q O R ) | 9 |

77 | Which one of the following statement is true for the given circle? A ( cdot overline{C D} cong overline{A B} ) B. ( overline{C D} neq overline{A B} ) c. ( widehat{C D} cong overline{A B} ) ( mathbf{D} cdot overline{C D} cong overline{A B} ) | 9 |

78 | A circle touches the hypotenuse of a right-angled triangle at its middle point and passes through the mid-point of the shorter side. If ( a ) and ( b(a<b) ) be the length of the sides, then prove that the radius is ( frac{b}{4 a} sqrt{a^{2}+b^{2}} ) | 10 |

79 | Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines. | 10 |

80 | What is ( angle D A E ) from the figure, if ( B C=D E=5 ) and ( angle B A C=45^{circ} ? ) A . ( 30^{circ} ) B . ( 45^{circ} ) ( c cdot 50^{circ} ) D. ( 60^{circ} ) | 9 |

81 | Tangents TP and TO are drawn from a point ( mathbf{T} ) to the circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=mathbf{a}^{2} . ) If the point ( mathbf{T} ) lies on the line ( mathbf{p x}+mathbf{q y}=mathbf{r} ) then the locus of centre of the circumcircle of ( Delta ) TPO is A ( cdot p x+q y=frac{r}{3} ) B. ( p x+q y=frac{r}{2} ) c. ( p x+q y=2 r ) D. ( p x+q y=r ) | 10 |

82 | Define the following term of the circle. Chord | 9 |

83 | In the given figure ‘O’ is the centre of the circle and ( A B, C D ) are equal to chords. If ( <A O B=70^{circ} k . ) Find the angles of ( triangle O C D ) | 9 |

84 | Two distinct chords drawn from the point ( (p, q) ) on the circle ( x^{2}+y^{2}= ) ( p x+q y ) are bisected at the ( x ) -axis. Then A ( cdot|p|=|q| ) B ( cdot p^{2}=8 q^{2} ) ( mathbf{c} cdot p^{2}8 q^{2} ) | 9 |

85 | Find the value of ( q ) ( A cdot 3 sqrt{5} ) B. ( 3 sqrt{10} ) ( c cdot 2 sqrt{10} ) D. ( 8 sqrt{10} ) | 10 |

86 | The angle between tangents drawn from the point (-1,3) to the circle ( x^{2}+ ) ( boldsymbol{y}^{2}=mathbf{5} ) is A. в. c. D. | 9 |

87 | Given, a circle with designated center designated perpendicular and radius 5 units. Find the length of the segment labeled ( boldsymbol{x} ) ( A cdot 4 ) B. 5 ( c cdot 8 ) D. ( sqrt{10} ) E ( . sqrt{3} ) | 9 |

88 | Find the value of ( J K ) in the following figure if ( angle boldsymbol{H} boldsymbol{L} boldsymbol{G}=angle boldsymbol{J} boldsymbol{L} boldsymbol{K} ) A . 11 3 ( c cdot 21 ) D. | 9 |

89 | What is chord? | 9 |

90 | A line that intersects a circle at two distinct points is called A . a diameter B. a secant c. a tangent D. a radius | 10 |

91 | The circumference of the circle is calculated by the formula ( mathbf{A} cdot 4 pi r ) В. ( 2 pi r^{2} ) c. ( 2 pi r ) D. ( pi r^{2} ) | 9 |

92 | A tangent ( P Q ) at a point ( P ) of a circle of radius ( 5 mathrm{cm} ) meets a line through the centre ( O ) so that ( O Q=13 mathrm{cm} . ) Find the length of ( P Q ) | 10 |

93 | The internal centre of similitude of two circles ( (x-3)^{2}+(y-2)^{2}= ) ( mathbf{9},(boldsymbol{x}+mathbf{5})^{2}+(boldsymbol{y}+mathbf{6})^{2}=mathbf{9} ) is A ( cdot(-1,-2) ) B. (-2,-1) c. (3,2) (年. (3,2) D. (-5,-6) | 9 |

94 | The perpendicular from the centre of a circle to a chord bisects the chord. | 9 |

95 | Find the radius of the circle which passes through the origin, (0,4) and (4,0) A. ( sqrt{8} ) B. 4 c. 16 D. ( sqrt{36} ) | 9 |

96 | Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60 | 10 |

97 | In the figure, the radius of the smaller circle is 3 centimetres, that of the bigger circle is 6 centimetres and the distance between the centres of the circles is 15 centimetres. PQ is a tangent to both the circles. Find its length. | 10 |

98 | Equal chords of a circle subtend equal angle on centre A . True B. False | 9 |

99 | Define congruent chords. | 9 |

100 | If the length of the largest chord of a circle is ( 17 mathrm{cm}, ) find the radius of a circle. ( A cdot 34 mathrm{cm} ) B. ( 8.5 mathrm{cm} ) c. ( sqrt{17} mathrm{cm} ) D. ( sqrt{34} mathrm{cm} ) | 9 |

101 | Draw the two tangents from a point which is ( 9 mathrm{cm} ) away from the centre of a circle of radius ( 3 mathrm{cm} ). Also, measure the lengths of the tangents. | 10 |

102 | ( f angle C=angle D=50^{circ}, ) then four points ( A, B ) ( C, D: ) A. Are con-cyclic B. Do not lie on same circle c. Are collinear A,B.D and A,B,C lie on different circles | 9 |

103 | The line ( x=y ) touches a circle at the point ( (1,1) . ) If the circle also passes through the point (1,-3) then its radius is: begin{tabular}{l} A ( .3 sqrt{2} ) \ hline end{tabular} B. 3 c. ( 2 sqrt{2} ) D. 2 | 10 |

104 | If the tangents ( P A ) and ( P B ) are drawn from the point ( mathbf{P}(-1,2) ) to the circles ( x^{2}+y^{2}+x-2 y-3=0 ) and ( C ) is the centre of the circle, then the area of the quadrilateral PACB is A .4 B. 16 c. does not exist D. 12 | 10 |

105 | In the diagram, ( A, B, C, D, E ) are points on the circle. ( A B | D C, angle A D E=39^{circ} ) and ( angle A B C=62^{circ} . ) Then the values of and ( y ) respectively are: A ( cdot 23^{circ}, 51^{circ} ) B ( cdot 79^{circ}, 62 ) ( c cdot 62^{circ}, 79^{circ} ) ( 0.51^{circ}, 23 ) | 9 |

106 | In the given figure, ( P A ) and ( P B ) are tangents from an external point ( boldsymbol{P} ) to a circle with center ( O . L N ) touches the circle at ( M . ) Prove that ( boldsymbol{P} boldsymbol{L}+boldsymbol{L} boldsymbol{M}= ) ( boldsymbol{P} boldsymbol{N}+boldsymbol{M} boldsymbol{N} ) | 10 |

107 | Find the diameter of the circle if its. Circumference is ( 62.8 mathrm{cm}(pi=3.14) ) | 9 |

108 | n Fig.1, 0 is the centre of circle, ( A B ) is a chord and ( A T ) is the tangent at ( A ). If ( angle A O B=100^{circ}, ) then ( angle B A T ) is equal to A ( cdot 100^{circ} ) B. ( 40^{circ} ) ( c cdot 50 ) D. 9 ? | 10 |

109 | The radius of the circle with centre at the origin is 10 units. Write the coordinates of the point where the circle intersects the axes. Find the distances between any two of such points. A ( . ) Co-ordinates ( =(10,0)(-10,0)(0,10)(0,-10) ) Distance ( =20,10 sqrt{2} ) units B. ( C o- ) ordinates ( =(10,0)(-10,0)(0,10)(0,-10) ) Distance ( =10,10 sqrt{2} ) units c. ( C o- ) ordinates ( =(10,0)(-10,0)(0,10)(0,-10), 0 ) Distance ( =20 sqrt{2} ) or ( 10 sqrt{2} ) units D. none | 9 |

110 | The lines ( 3 x+4 y=9 ) and ( 6 x+8 y+ ) ( mathbf{1 5}=mathbf{0} ) are tangents to the same circle. The radius of the circle is :- A ( cdot frac{3}{10} ) в. ( frac{33}{20} ) ( c cdot frac{33}{10} ) D. ( frac{33}{5} ) | 10 |

111 | In the adjoining figure ( A O B ) is a diameter ( M P Q ) is a tangent at ( P ) then the value of ( angle M P A ) is equal to A ( cdot 25 ) B ( .26^{circ} ) ( c cdot 27^{circ} ) ( D cdot 30^{circ} ) | 10 |

112 | A pair of opposite sides of a cyclic quadrilateral are equal. Prove that its diagonal are also equal | 9 |

113 | Two parallel chords are drawn in a circle of diameter ( 30.0 mathrm{cm} . ) The length of one chord is ( 24.0 mathrm{cm} ) and the distance between the two chords is ( 21.0 mathrm{cm} . ) find the length of the other chord. | 9 |

114 | Tangents ( P A ) and ( P B ) are drawn from an external point ( P ) to two concentric circle with centre ( O ) and radii ( 8 mathrm{cm} ) and 5 ( mathrm{cm} ) respectively, as shown in figure. If ( A P=15 mathrm{cm}, ) then find the length of ( boldsymbol{B P} ) | 10 |

115 | CP and ( mathrm{CQ} ) are tangents to a circle with centre ( 0 . A R B ) is another tangent touching the circle at ( mathrm{R} . ) If ( C P= ) ( 11 c m, B C=7 c m, ) then the length BR is ( A cdot 11 c m ) B. ( 7 mathrm{cm} ) ( c .3 c m ) ( mathrm{D} cdot 4 mathrm{cm} ) | 10 |

116 | Through a fixed point ( (h, k) ) secants are drawn to the circle ( x^{2}+y^{2}=r^{2} ) Then the locus of the midpoints of the chords intercepted by the circle is A ( cdot x^{2}+y^{2}=h x+k y ) B . ( x^{2}-y^{2}=h x+k y ) C. ( x^{2}+y^{2}=h x-k y ) D. ( x^{2}-y^{2}=h x-k y ) | 10 |

117 | In a circle with center ( O, ) a chord ( P Q ) is such that ( boldsymbol{O} boldsymbol{M} pm boldsymbol{P} boldsymbol{Q} ) meeting ( boldsymbol{P} boldsymbol{Q} ) at ( boldsymbol{M} ) Then ( ^{mathbf{A}} cdot O Q^{2}=O M^{2}+frac{1}{2} P Q^{2} ) B. ( O Q^{2}=O M^{2}+frac{1}{4} P Q^{2} ) c. ( M Q^{2}=O M^{2}-O Q^{2} ) D. ( O M^{2}=M Q^{2}-O Q^{2} ) | 9 |

118 | If the squares of the lengths of the tangents from a point ( P ) to the circles ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{a}^{2}, boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{b}^{2} ) and ( boldsymbol{x}^{2}+ ) ( y^{2}=c^{2} operatorname{are} ) in A.P., then This question has multiple correct options A ( cdot a^{2}, b^{2}, c^{2} ) are in A.P B. ( frac{1}{a^{2}}, frac{1}{b^{2}}, frac{1}{c^{2}} ) are in ( mathrm{H.P} ) c. ( a^{2}, b^{2}, c^{2} ) are in G.P D. ( frac{1}{a^{2}}, frac{1}{b^{2}}, frac{1}{c^{2}} ) are in A.F. | 10 |

119 | statement-l: rrom a poınt ( boldsymbol{r} ) on tne circle with centre ( boldsymbol{O} ) the chord ( boldsymbol{P} boldsymbol{A}=mathbf{8} ) ( mathrm{cm} ) is drawn. The radius of the circle is ( 24 mathrm{cm} . ) Let ( P B ) be drawn parallel to ( O A ) Suppose ( B O ) extended meet ( P A ) extended at ( M . ) The length of ( M A ) is 9 ( mathrm{cm} ) Reason Statement-2: ( O A ) is a radius of a circle with centre at ( O . R ) is a point on ( O A ) through which a chord ( C D ) perpendicular to ( boldsymbol{O} boldsymbol{A} ) is drawn. Let a chord through A meet the chord ( C D ) at ( M ) and the circle at ( B ). Also ( O S ) is perpendicular from ( boldsymbol{O} ) on chord ( boldsymbol{A} boldsymbol{B} ). The radius of the circle is ( 18 mathrm{cm} . R ) is the mid point of ( boldsymbol{A O} ) and ( boldsymbol{A} boldsymbol{M} / boldsymbol{M} boldsymbol{B}=frac{mathbf{1}}{mathbf{2}} ) The length of ( boldsymbol{O} boldsymbol{S} ) is ( boldsymbol{9} mathrm{cm} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 9 |

120 | The radius of a circle is ( 13 mathrm{cm} ) and the length of one of its chords is ( 10 mathrm{cm} ). The distance of the chord from the centre is A. ( 8 mathrm{cm} ) B. ( 10 mathrm{cm} ) ( mathrm{c} cdot 12 mathrm{cm} ) D. ( 6 mathrm{cm} ) | 9 |

121 | In Fig. 3 , if 0 is the centre of the circle ( mathrm{OL}=4 mathrm{cm}, mathrm{AB}=6 mathrm{cm} ) and ( mathrm{OM}=3 ) ( mathrm{cm}, ) then ( mathrm{CD}=? ) ( A cdot 4 c m ) в. ( 8 mathrm{cm} ) ( c . ) 6cm D. 10cm | 9 |

122 | n fig, chord ( A B | C D ) of a circle with centre ( O ) and radius 5 cm such that ( A B=6 mathrm{cm} ) and ( C D=8 mathrm{cm} . ) if ( O P ) ( perp A B, O Q . perp C D, ) then ( P Q ) in cm is ( mathbf{A} cdot 4 mathrm{cm} ) ( mathbf{B} cdot 7 mathrm{cm} ) ( c cdot 1 mathrm{cm} ) D. ( 3 mathrm{cm} ) | 9 |

123 | Find the centre and radius of the circle ( x^{2}+y^{2}-4 x-8 y-45=0 ) | 9 |

124 | In the given diagram, ( O ) is the centre of the circle and ( P Q R ) is a straight line. The value of ( x ) is A ( cdot 110^{circ} ) B. ( 120^{circ} ) ( c cdot 130^{circ} ) D. ( 140^{circ} ) | 9 |

125 | Recall that two circles are congruent if they have the same radius then equal chords of congruent circles subtend equal angles at their centres. A . True B. False | 9 |

126 | Each of the height and radius of the base of a right circular cone is increased by ( 100 % ). The volume of the cone will be increased by A . ( 700 % ) в. ( 500 % ) c. ( 300 % ) D. ( 100 % ) | 9 |

127 | ( A B C D ) is a cyclic quadrilateral such that ( A B ) is a diameter of the circle circumscribing it and ( angle A D C=140^{circ} ) then ( angle B A C ) is equal to ( A cdot 80 ) B. ( 50^{circ} ) ( c cdot 40^{circ} ) D. ( 30^{circ} ) | 9 |

128 | A circular area having a radius ( 20 mathrm{cm} ) is divided into two equal parts by a concentric circle of radius ‘r’. The value of ‘r’ will be A. ( 5 mathrm{cm} ) B. 10 ( mathrm{cm} ) ( mathrm{c} cdot 5 sqrt{2} mathrm{cm} ) D. ( 10 sqrt{2} mathrm{cm} ) | 9 |

129 | 69. The circumcentre of a triangle ABC is O. If Z BAC = 85° and BCA = 75°, then the value of 2 OAC is (1) 40° (2) 60° (3) 70° (4) 90° | 9 |

130 | Line segment joining the centre to any point on the circle is A. radius of the circle B. diameter of the circle c. secant of the circle D. tangent of the circle. | 9 |

131 | In the given figure points ( A, D, P, C ) and ( B ) lie on a circle with centre ( boldsymbol{O}, angle boldsymbol{B O D}=mathbf{1 5 0}^{circ} ) Find the measures of ( angle B P D, angle B C D ) and ( angle B A D ) | 9 |

132 | Equation of a straight line meeting the circle ( x^{2}+y^{2}=100 ) in two points each point at a distance of 4 from the point (8,6) on the circle is A. ( 4 x+3 y-50=0 ) B. ( 4 x+3 y-100=0 ) c. ( 4 x+3 y-46=0 ) D. none of these | 9 |

133 | If the line ( h x+k y=1 ) touches ( x^{2}+ ) ( y^{2}=a^{2}, ) then the locus of the point ( (h ) k) is a circle of radius ( A cdot a ) B. ( c cdot sqrt{a} ) D. ( frac{1}{sqrt{a}} ) | 9 |

134 | what is tangent of a circle and definition? | 10 |

135 | If the common chord of the circle ( x^{2}+ ) ( (y-lambda)^{2}=16 ) and ( x^{2}+y^{2}=16 ) subtend a right angle at the origin then ( lambda ) is equal to ( A cdot 4 ) B. ( 4 sqrt{2} ) ( c cdot pm 4 sqrt{2} ) ( D .8 ) | 9 |

136 | A and ( mathrm{B} ) are two points on the circle ( mathbf{x}^{2}+mathbf{y}^{2}=1 . ) If the ( mathbf{x} ) co-ordinates of ( mathbf{A} ) and ( mathrm{B} ) are the roots of the equation ( x^{2}+a x+b=0 ) and the ( y ) coordinates of ( mathbf{A} ) and ( mathbf{B} ) are the roots of the equation ( mathbf{y}^{2}+mathbf{b y}+mathbf{a}=mathbf{0} ) then the equation of the line ( A B ) is A ( cdot a x+b y=0 ) B. ( a x+b y+1=0 ) c. ( b x+a y+a+b=0 ) D. ( a x+b y+a+b+1=0 ) | 9 |

137 | The tangents drawn at the ends of a diameter of a circle are ? A. perpendicular B. parallel c. adjacent D. none of the above | 10 |

138 | In the given figure, ( Delta X Y Z ) is inscribed in a circle with centre 0. If the length of chord YZ is equal to the radius of the circle OY then ( angle boldsymbol{Y} boldsymbol{X} boldsymbol{Z}= ) A ( cdot 60^{circ} ) B. ( 30^{circ} ) ( c cdot 80^{circ} ) D. ( 100^{circ} ) | 9 |

139 | In figure, ( boldsymbol{K} boldsymbol{X} boldsymbol{M} ) is a tangent to the circumcircle ( C ) of ( triangle X Y Z ) such that ( boldsymbol{L} boldsymbol{M} | boldsymbol{Y} boldsymbol{Z} . ) Show that ( boldsymbol{X} boldsymbol{Y}=boldsymbol{X} boldsymbol{Z} ) | 9 |

140 | In the figure, 0 is the centre of the circle Find the length of ( mathrm{CD} ), if ( mathrm{AB}=5 mathrm{cm} ) | 9 |

141 | 69. O is the circumcentre of A ABC. If Z BAC = 85°, Z BCA = 75°, then 2 OAC is equal to (1) 70° (2) 60° (3) 80° (4) 100° | 9 |

142 | Assertion The circle of smallest radius passing through two given points ( A & B ) must be of radius ( frac{1}{2} A B ) Reason A straight line is a shortest distance between two points. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 9 |

143 | The square of the length of the tangent from (3,-4) to the circle ( x^{2}+y^{2}- ) ( 4 x-6 y+3=0 ) is A . 20 B. 30 ( c cdot 40 ) D. 50 | 10 |

144 | For each ( k epsilon N, ) let ( C_{k} ) denote the circle whose equation is ( x^{2}+y^{2}=k^{2} . ) On the circle ( C_{k}, ) a particle moves k units in the anticlockwise direction. After completing its motion on ( C_{k}, ) the particle moves to ( C_{k+1} ) in the radial direction. The motion of the particle continues in this manner. The particle starts at ( (1,0) . ) If the particle crossed the positive direction of the ( x ) -axis for the first time of the circle ( C_{n} ) then ( n ) is A . 7 B. 6 ( c cdot 2 ) D. none of these | 9 |

145 | In the figure, if ( boldsymbol{O A}=mathbf{1 7 c m}, boldsymbol{A B}= ) ( 30 mathrm{cm} ) and ( 0 mathrm{D} ) is perpendicular to ( mathrm{AB} ) then ( mathrm{CD} ) is equal to: ( A cdot 8 c m ) в. ( 9 mathrm{cm} ) ( c cdot 10 c m ) D. ( 11 mathrm{cm} ) | 9 |

146 | If ( bar{P} Q ) is a chord of a circle with centre ( O ) and ( P R ) is a tangent to the circle at ( P ) then ( angle P O Q= ) A. ( 4 angle R P Q ) B. ( 3 angle R P Q ) c. ( 2 angle R P Q ) D. ( angle R P Q ) | 9 |

147 | State the following statement is True or False If the chords of a circle intersect within | 9 |

148 | From the following figure, choose the statements that are correct. i) Congruent chords have congruent ( operatorname{arcs} ) ii) Congruent chords have equal centra angles. iii) Congruent arcs have congruent central angles. iv) Chords equidistant from the center are congruent. A. ii and iii only B. iii and iv D. All of the above | 9 |

149 | Show that all the chords of the curve ( 3 x^{2}-y^{2}-2 x+4 y=0 ) which subtend a right angle at the origin? | 9 |

150 | The diameter of the circle is ( 2 mathrm{cm} ). What is the circumference? ( mathbf{A} cdot 12.28 mathrm{cm} ) B. ( 6.2 mathrm{cm} ) c. ( 18.28 mathrm{cm} ) D. ( 10.28 mathrm{cm} ) | 9 |

151 | In a circle if a chord ( A B ) is nearer to the centre ( boldsymbol{O} ) than the chord ( boldsymbol{C} boldsymbol{D} ) then: ( mathbf{A} cdot A B>C D ) B. ( A B=C D ) c. ( A B<C D ) D. none of these | 9 |

152 | Define diameter. | 9 |

153 | Prove that the centre of the smallest circle passing through origin and whose centre lies on ( boldsymbol{y}=boldsymbol{x}+mathbf{1} ) is ( left(-frac{mathbf{1}}{mathbf{2}}, frac{mathbf{1}}{mathbf{2}}right) ) | 9 |

154 | 68. The length of a chord of a circle is equal to the radius of the cir- cle. The angle which this chord subtends in the major segment of the circle is equal to (1) 30° (2) 45° (3) 60° (4) 90° | 9 |

155 | In the figure, ( boldsymbol{O} ) is the point of intersection of two chords ( A B ) and ( C D ) such that ( O B=O D ), then triangles ( O A C ) and ( O D B ) are: A. Equilateral but not similar B. Isosceles but not similar c. Equilateral and similar D. Isosceles and similar | 9 |

156 | A circle touches the sides of a quadrilatieral ABCD at P, Q, R, S respectively. The angles subtended at the centre by a pair of opposite sides have theirs sum as: | 10 |

157 | Chords of the circle ( x^{2}+y^{2}+2 g x+ ) ( 2 f y+c=0 ) subtends a right angle at the origin. The locus of the feet of the perpendiculars from the origin to these chords is A ( cdot x^{2}+y^{2}+g x+f y+c=0 ) B . ( 2left(mathrm{x}^{2}+mathrm{y}^{2}right)+mathrm{gx}+mathrm{fy}+mathrm{c}=0 ) C ( cdot 2left(mathrm{x}^{2}+mathrm{y}^{2}+mathrm{gx}+mathrm{fy}right)+mathrm{c}=0 ) D. ( x^{2}+y^{2}+2(g x+f y+c)=0 ) | 9 |

158 | Find the centres of the circles passing through (-4,3) and touching the lines ( boldsymbol{x}+boldsymbol{y}=boldsymbol{2} ) and ( boldsymbol{x}-boldsymbol{y}=boldsymbol{2} ) A ( cdot((-10 pm sqrt{54}), 0) ) B. ( (10 pm sqrt{54}, 0) ) c. ( (0,-10 pm sqrt{54}) ) D. ( (0,10 pm sqrt{54}) ) | 10 |

159 | Find the length of a chord which is at a distance of ( 3 mathrm{cm} ) from the centre of a circle of radius ( 5 mathrm{cm} ) A . ( 2 mathrm{cm} ) B. ( 6 mathrm{cm} ) ( c cdot 8 mathrm{cm} ) D. ( 10 mathrm{cm} ) | 9 |

160 | If ( angle R P Q=45^{circ}, ) then find ( angle P Q R ) ( mathbf{A} cdot 15^{circ} ) B ( .30^{circ} ) ( c cdot 60^{circ} ) D ( .45^{circ} ) | 10 |

161 | Prove that if chords of congruent circles subtend equal angle at their centres, then the chords are equal. | 9 |

162 | Find the value of ( x+y ) in the given figure (in degrees) | 10 |

163 | From the figure, identify a sector | 9 |

164 | Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres. | 9 |

165 | In a circle of diameter ( 10 mathrm{cm} ), the length of each of 2 equal and parallel chords is ( 8 mathrm{cm}, ) then the distance between these two chords is A. ( 4 mathrm{cm} ) B. ( 5 mathrm{cm} ) ( c cdot 6 mathrm{cm} ) D. ( 7 mathrm{cm} ) | 9 |

166 | Draw any circle and mark an arc. | 9 |

167 | Find the distance of a perpendicular from the centre of a circle to the chord if the diameter of the circle is ( 30 mathrm{cm} ) and its chord is ( 24 mathrm{cm} ) A ( .6 mathrm{cm} ) B. ( 7 mathrm{cm} ) ( c .9 mathrm{cm} ) D. ( 10 mathrm{cm} ) | 9 |

168 | If the chord ( y=m x+1 ) of the circle ( x^{2}+y^{2}=1 ) subtends an angle of measure ( 45^{circ} ) at the major segment of the circle then the value of ( m ) is A ( .2 pm sqrt{2} ) B. ( -2 pm sqrt{2} ) c. ( -1 pm sqrt{2} ) D. none of these | 9 |

169 | In given figure, ( P Q ) is chord of length ( 8 c m ) of a circle of radius ( 5 c m, ) the tangents at ( P ) and ( Q ) intersect at a point T. Find the length ( boldsymbol{T} boldsymbol{P} ) | 9 |

170 | A secant intersects the circle at points. ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) ( D ) | 10 |

171 | Tangents are drawn to the circle ( x^{2}+ ) ( y^{2}=25 ) from the point ( (13,0) . ) Prove that the angle between them is ( 2 tan ^{-1}(5 / 12) ) and their equations are ( 12 y+5 x+65=0 ) and ( 12 y-5 x- ) ( mathbf{6 5}=mathbf{0} ) | 10 |

172 | 66. Two chords AB and CD of a cir- cle with centre O intersect each other at the point P. If ZAOD = 20° and ZBOC = 30°, then ZBPC is equal to: (1) 50° (2) 20° (3) 25° (4) 30° | 9 |

173 | The length of a chord of a circle of radius ( 10 mathrm{cm} ) is ( 12 mathrm{cm} ). Find the distance of the chord from the centre of the circle A. ( 6 mathrm{cm} ) B. ( 5 mathrm{cm} ) ( c .8 mathrm{cm} ) D. ( 7 mathrm{cm} ) | 9 |

174 | Find the equation of the tangent to the curve ( y=x^{2}-7 ) at the point where it cuts the ( y ) – axis. | 10 |

175 | Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at a center. | 10 |

176 | The distance between two parallel chords, each of length 10 units is 24 units then the radius of the circle is: A . 5 B. 12 c. 13 D. 30 | 9 |

177 | ( O ) is the centre of the circle. ( A B ) and ( C D ) are two chords of the circle. ( O M perp ) ( A B ) and ( O N perp C D . ) If ( O M=O N=3 ) ( operatorname{cmand} A M=B M=4.5 mathrm{cm}, ) then ( C D ) is equal to ( A cdot 8 mathrm{cm} ) B. 9 cm ( c .10 mathrm{cm} ) D. None of these | 9 |

178 | In the given figure, 0 is the centre of the circle and XOY is a diameter. If ( X Z ) is any other chord of the circle, then which of the following is correct? ( A cdot X Zx z ) ( c cdot 0 x+0 z ) D. zX + ZY | 9 |

179 | A boat in a circular lake lies at its centre. The perpendicular distance of the boat is ( 10 mathrm{m} ) from a bridge lying in ( 40 mathrm{m} ) distance across the circular lake. Find the distance that the boat will have to travel to reach to the extreme point of left side of bridge is ( m sqrt{5} mathrm{m} ). Then, ( m ) is A . ( 20 mathrm{m} ) в. ( 10 mathrm{m} ) ( c .-10 m ) D. both B & | 9 |

180 | Find the center and radius of the circle. ( (x+5)^{2}+(y-3)^{2}=36 ) | 9 |

181 | What is the angle between the line joining the centre and point of contact of a tangent and the tangent itself? ( mathbf{A} cdot mathbf{0} ) B . 45 ( c .90 ) D. ( 180^{circ} ) | 10 |

182 | The radius of a circle is ( 40 mathrm{cm} ) and the length of perpendicular drawn from its centre to chord is ( 24 mathrm{cm} . ) The length of the chord ( A B ) is A. ( 32 mathrm{cm} ) B. 64cm c. ( 48 mathrm{cm} ) D. 24cm | 9 |

183 | ( l x+m y+n=0 ) is a tangent line to the circle ( x^{2}+y^{2}=r^{2}, ) if A ( cdot l^{2}+m^{2}=n^{2} r^{2} ) B . ( l^{2}+m^{2}=n^{2}+r^{2} ) C ( cdot n^{2}=r^{2}left(l^{2}+m^{2}right) ) D. none of these | 10 |

184 | circle is a 69. Two cu Two circles of diameters 10 and 6 cm have the same cene A chord of the larger circle tangent of the smaller one, length of the chord is (1) 4 cm. (2) 8 cm. (3) 6 cm. (4) 10 cm | 9 |

185 | 62. The diagonals AC and BD of a cyclic quadrilateral ABCD inter- sect each other at the point P. Then, it is always true that (1) BP. AB = CD. CP (2) AP. CP = BP. DP (3) AP.BP = CP. DP (4) AP. CD = AB .CP | 9 |

186 | A point ( P ) is outside a circle at a distance of ( 13 mathrm{cm} ) from its centre. secant from ( P ) cuts the circle in ( Q ) and ( R ) such that ( Q R=7 mathrm{cm} ) and the segment ( P Q ) of the secant, exterior to the circle is ( 9 mathrm{cm} . ) Therefore, the radius of circle is A. 3 cm в. 4 ст ( c .5 mathrm{cm} ) D. ( 6 mathrm{cm} ) | 9 |

187 | In the given figure, ( A B ) is a chord of length ( 16 mathrm{cm} ) of a radius ( 10 mathrm{cm} . ) The tangents at ( A ) and ( B ) intersect at point ( P . ) Find the length of ( P A ) | 10 |

188 | The equation of the circle and its chord are respectively ( x^{2}+y^{2}=a^{2} ) are ( x cos alpha+y sin alpha=p . ) The equation of the circle of which this chord is diameter is A ( cdot x^{2}+y^{2}-2 p x cos alpha-2 p y sin alpha+2 p^{2}-a^{2}=0 ) B . ( x^{2}+y^{2}-2 p x cos alpha-2 p y sin alpha+p^{2}-a^{2}=0 ) C ( cdot x^{2}+y^{2}-2 p x cos alpha+2 p y sin alpha+2 p^{2}-a^{2}=0 ) D. None of these | 9 |

189 | A point ( P ) is ( 13 mathrm{cm} ) from the centre of the circle. The length of the tangent drawn from ( P ) to the circle is 12 cm. Find the radius of the circle. | 10 |

190 | Write True or False and justify your answer in each of the following: If a number of circles touch a given line segment PQ at a point ( A, ) then their centres lie on the perpendicular bisector of PQ. A. True B. False c. Ambiguous D. Data Insufficient | 9 |

191 | Illustration 2.21 Find the length of an arc of a circle of radius 5 cm subtending a central angle measuring 15º. | 9 |

192 | n Fig. 0 is the centre of the circle such that ( angle A O C=130^{circ}, ) then ( angle A B C= ) A ( cdot 130 ) B. 115 ( c cdot 65 ) D. 165 | 9 |

193 | Two parallel chords in a circle are ( 10 mathrm{cm} 10 mathrm{cm} ) and ( 24 mathrm{cm} 24 mathrm{cm} ) long. If the radius of the circle is ( 13 mathrm{cm} 13 mathrm{cm} ) find the distance between the chords if thay lie on the same side of the centre. | 9 |

194 | n Fig. ( A B ) and ( C D ) are common tangents to two circles of unequal radii then ( A B ) is not equal to ( mathrm{CD} ) A. True 3. Falss | 10 |

195 | STATEMENT – 1: The locus of the middle points of equal chords of a circle with centre at 0 is a circle with centre at 0 STATEMENT – 2 : The mid point of the equal chords are equidistant from the centre of the circle. A. Statement – 1 is True, Statement – 2 is True, Statement 2 is a correct explanation for Statement – 1 B. Statement – 1 is True, Statement – 2 is True : Statement 2 is NOT a correct explanation for Statement- – c. Statement – 1 is True, Statement – 2 is False D. Statement-1 is False, Statement- – 2 is True | 9 |

196 | A tangent is drawn to each of the circles ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{a}^{2}, boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{b}^{2} ) Show that if the two tangents are mutually perpendicular, the locus of their point of intersection is a circle concentric with the given circles. | 10 |

197 | ( A B ) and ( C D ) are two parallel chords of a circle of radius ( 3 mathrm{cm} . ) If ( A B=4 mathrm{cms} ) and ( C D=5 mathrm{cm} . ) Then the distance between them (in ( mathrm{cm} ) ) is A ( cdot frac{sqrt{5}}{2}+sqrt{11} ) B. ( sqrt{5}+sqrt{11} ) ( ^{c} cdot sqrt{5}+frac{sqrt{11}}{2} ) D. ( sqrt{2}+frac{sqrt{11}}{sqrt{5}} ) | 9 |

198 | Find the length of a chord which is at a distance of ( 4 mathrm{cm} ) from the centre of a circle whose radius is ( 5 mathrm{cm} ) | 9 |

199 | The condition that the chord ( x cos alpha+ ) ( boldsymbol{y} sin boldsymbol{alpha}-boldsymbol{p}=mathbf{0} ) of ( boldsymbol{x}^{2}+boldsymbol{y}^{2}-boldsymbol{a}^{2}=mathbf{0} ) may subtend a right angle at the centre of the circle is A ( cdot a^{2}=2 p^{2} ) B ( cdot p^{2}=2 a^{2} ) c. ( a=2 p ) D. ( p=2 a ) | 9 |

200 | Three circles with centre ( A, B ) and ( C ) respectively, touch one another as shown in the figure. If ( A, B ) and ( C ) are collinear and PQ is a common tangent to the two smaller circles, where ( mathrm{PQ}=4 ) the area of shaded region is | 10 |

201 | Find the centre and radius of the circle ( 2 x^{2}+2 y^{2}=3 x-5 y+7 ) | 9 |

202 | Out of the two concentric circles, the radius of the outer circle is ( 5 mathrm{cm} ) and the chord ( A C ) of length ( 8 mathrm{cm} ) is a tangent to the inner circle. Find the radius of the inner circle. ( A .3 c m ) B. ( 6 mathrm{cm} ) ( c .5 mathrm{cm} ) D. 7 cm | 9 |

203 | The center of a circle which passes through the points (0,0),(1,0) and touches the circles ( x^{2}+y^{2}=9 ) ( ^{A} cdotleft(frac{3}{2}, frac{1}{2}right) ) в. ( left(frac{1}{2}, frac{3}{2}right) ) c. ( left(frac{1}{2}, frac{1}{2}right) ) D. ( left(frac{1}{2}, sqrt{2}right) ) | 9 |

204 | What are the coordinates of the center of the circle represented by the equation ( (x+3)^{2}+(y-4)^{2}=25 ? ) ( A cdot(3,4) ) B. (3,-4) c. (-3,4) D. (-3,-4) | 9 |

205 | If the area and the circumference of circle are numerically equal, then the radius the circle is ( ^{A} cdot frac{5}{2} ) B. 2 c. 1 D. 2 ( overline{5} ) | 9 |

206 | 56. Each of the circles of equal radil with centres A and B pass through the centre of one anoth- er circle they cut at C and D then DBC is equal to (1) 60° (2) 100 (3) 120° (4) 140° | 9 |

207 | The common chord of ( x^{2}+y^{2}-4 x- ) ( 4 y=0 ) and ( x^{2}+y^{2}=16 ) subtends at the origin an angle to A. ( pi / 6 ) в. ( pi / 4 ) c. ( pi / 3 ) D. ( pi / 5 ) | 9 |

208 | ( operatorname{Let} O P=5 ) and ( P M=4 ) Find ( O M ) ( A cdot 3 c m ) B. ( 4 mathrm{cm} ) ( c .5 mathrm{cm} ) D. ( 8 c m ) | 9 |

209 | If radii of two concentric circles are 4 ( mathrm{cm} ) and ( 5 mathrm{cm}, ) then the length of each chord of one circle which is tangent to the other circle is A. ( 3 mathrm{cm} ) B. 6 ( mathrm{cm} ) ( c cdot 9 mathrm{cm} ) D. ( 1 mathrm{cm} ) | 10 |

210 | 69. The length of a tangent from an external point to a circle is 5/3 unit. If radius of the circle is 5 units, then the distance of the point from the circle is (1) 5 units (2) 15 units (3) -5 units (4) -15 units | 10 |

211 | Prove that the tangents drawn from an external point to a circle are equal. | 10 |

212 | In the figure, ( M N S ) is tangent to the circle with centre ( O ) at ( N ) ( A B ) is chord parallel to ( M N S . ) find ( angle A N C ) A . 50 B. ( 90^{circ} ) ( c cdot 40 ) D. 20 | 9 |

213 | The equation of the circle, passing through the point ( (2,8), ) touching the ( operatorname{lines} 4 x-3 y-24=0 ) and ( 4 x-3 y- ) ( 42=0 ) and having ( x ) coordinate of the center of the circle numerically less then or equal to 8 , is A ( cdot x^{2}+y^{2}+4 x-6 y-12=0 ) B. ( x^{2}+y^{2}-4 x+6 y-12=0 ) C ( cdot x^{2}+y^{2}-4 x-6 y-12=0 ) D. None of these | 10 |

214 | The longest chord passes through a centre of a circle is | 9 |

215 | In the diagram, ( P ) is the centre of the circle with radius ( 4 mathrm{cm} ) and ( Q ) is the centre of the circle with radius ( 3 mathrm{cm} ) Of the points marked ( W, X, Y ) and ( Z ) which point is ( 4 mathrm{cm} ) from ( P ) and more than ( 3 mathrm{cm} ) from ( Q ? ) ( A cdot W ) в. ( x ) ( c . Y ) D. | 9 |

216 | 75. In the given figure, POg is a di- ameter and PQRS is a cyclic quadrilateral. If ZPSR = 130°, then the value of ZRPO is 130° (1) 30° (3) 45° (2) 40° (4) 35° | 9 |

217 | In a circle with centre ( 0 . operatorname{seq} mathrm{PQ}, ) is a chord such that ( angle P O Q=70^{circ} . ) Find the ( angle O P Q ) | 9 |

218 | Suppose you are given a circle. Give steps of construction to find its centre. | 9 |

219 | n Fig. ( 2, ) ‘O’ is the centre of the circle, find ( angle A O C, operatorname{given} angle B A O=30^{circ} ) and ( angle B C O=40^{circ} ) ( A cdot 35 ) В. 140 ( c cdot 70 ) D. Cannot be determined | 9 |

220 | In the given figure, a circle with centre ( O ) is given in which a diameter ( A B ) bisects the chord ( C D ) at a point ( E ) such that ( C E=E D=8 c m ) and ( E B=4 c m ) Find the radius of the circle. | 9 |

221 | ACB is a tangent to a circle at c. CD and CE are chords such that ZACE > ZACD. If ZACD = ZBCE = 50°. then : (1) CD = CE (2) ED is not parallel to AB (3) ED passes through the cen- tre of the circle (4) A CDE is a right angled trian- gle | 9 |

222 | Find the value of ( c ) if (2,3) lies on the circle ( x^{2}+y^{2}+2 x+3 y+c=0 ) | 9 |

223 | Prove: If a chord of circle ( x^{2}+y^{2}=8 ) makes equal intercepts of length ‘a’ on the coordinate axes then ( |a|<4 ) | 9 |

224 | If ( O ) is the centre of a circle, ( P Q ) is a chord and the tangent ( P R ) at ( P ) makes an angle of ( 50^{0} ) with ( P Q ), then find the Angle ( (P O Q) ) | 10 |

225 | The tangents are drawn from origin and the point ( (boldsymbol{g}, boldsymbol{f}) ) to the circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+ ) ( 2 g x+2 f y+c=0 . ) Find the distance between chords of contact. A ( cdot frac{2left(g^{2}+f^{2}-cright)}{sqrt{g^{2}+f^{2}}} ) B. ( frac{g^{2}+f^{2}-c}{sqrt{g^{2}+f^{2}}} ) c. ( frac{g^{2}+f^{2}-c}{2 sqrt{g^{2}+f^{2}}} ) D. none of these | 9 |

226 | The length of a tangent from a point ( boldsymbol{A} ) at distance ( 5 mathrm{cm} ) from the centre of the circle is ( 4 mathrm{cm} ).Find the radius of the circle. | 9 |

227 | O’ is the centre of the circle ( angle Q P S= ) ( mathbf{6 5}^{circ} ; angle boldsymbol{P} boldsymbol{R} boldsymbol{S}=mathbf{3 3}^{circ}, )then ( angle boldsymbol{P S Q}= ) A .90 B. ( 82^{circ} ) ( c cdot 102 ) ( D cdot 42 ) | 9 |

228 | Find the equations to the circles in which the line joining the points ( (a, b) ) and ( (b,-a) ) is a chord subtending an angle of ( 45^{circ} ) at any point on its circumference. | 9 |

229 | Prove that the parallelogram circumscribing a circle is a rhombus. | 10 |

230 | If the points (2,0),(0,1),(4,5) and ( (0, c) ) are concyclic, then the value of ( c ) is This question has multiple correct options A . -1 B. c. ( frac{14}{3} ) D. ( -frac{14}{3} ) | 9 |

231 | In the given figure below, ( A D ) is a diameter. ( O ) is the centre of the circle. ( A D ) is parallel to ( B C ) and ( angle C B D=32^{circ} ) Find ( angle B E Dleft(text { in }^{circ}right) ) | 9 |

232 | 73. PO is a tangent to the circle at R then mZPRS is equal to : BOT (1) 30° (3) 60° (2) 40° (4) 80° | 9 |

233 | The equation of the circle with center (1,2) and tangent ( x+y-5=0 ) is A ( cdot x^{2}+y^{2}+2 x-4 y+6=0 ) B . ( x^{2}+y^{2}-2 x-4 y+3=0 ) c. ( x^{2}+y^{2}-2 x-4 y-8=0 ) D. ( x^{2}+y^{2}-2 x-4 y+8=0 ) | 10 |

234 | In the diagram, PQ and QR are tangents to the circle, centre ( 0, ) at ( P ) and ( R ) respectively. Find the value ( A cdot 25 ) 3.35 ( c cdot 45 ) 55 | 10 |

235 | Chord ( A B ) of the circle ( x^{2}+y^{2}=100 ) passes through the point (7,1) and subtends an angle of ( 60^{circ} ) at the circumference of the circle. If ( m_{1} ) and ( m_{2} ) are the slopes of two such chords then the value of ( m_{1} m_{2} ) is A . -1 B. c. ( 7 / 12 ) D. – | 9 |

236 | A chord of a circle of radius ( 12 mathrm{cm} ) subtends an angle of ( 120^{0} ) at the centre. Find the area of the corresponding segment | 9 |

237 | Find the radius of the circle passing the points (0,0),(1,0) and (0,1) | 9 |

238 | Tangent ( 0 A ) and ( O B ) are drawn for ( O(0,0) ) to the circle ( (x-1)^{2}+(y- ) 1) ( ^{2}=1 ) Equation of the circumcircle of triangle OAB is A ( cdot x^{2}+y^{2}+x+y=0 ) B . ( x^{2}+y^{2}-x+y=0 ) c. ( x^{2}+y^{2}+x-y=0 ) D. ( x^{2}+y^{2}-x-y=0 ) | 10 |

239 | The coordinates of the fixed point of the chord cut off by ( 2 x-5 y+18=0 ) by the circle ( x^{2}+y^{2}-6 x+2 y-54=0 ) are A ( .(1,4) ) в. (2,4) c. (4,1) () D. (1,1) | 9 |

240 | 59. Two circle with centres O and O’ touch externally each other at point P. A straight line is drawn from P which intersects both the circles at Q and R. Given that radii of the circles OP= 6 cm and O’P=4 cm and PQ = 10 cm, then PR = ? (1) 7.6 cm (2) 7.8 cm (3) 6.7 cm (4) 7.5 cm | 10 |

241 | If ( P ) is a point, then how many tangents to a circle can be drawn from the point ( P, ) if it lies On the circle. ( A cdot 0 ) B. ( c cdot 2 ) D. 3 | 10 |

242 | 71. The tangents at two points A and B on the circle with cen- tre O intersect at P: if in quadrilateral PAOB, ZAOB: ZAPB = 5:1, then measure of ZAPB is : (1) 30º (2) 60° (3) 45° (4) 15° | 9 |

243 | Circumference of a circle is equal to A . ( pi r ) в. ( 2 pi r ) c. ( frac{pi r}{2} ) D. ( 2+frac{pi r}{2} r ) | 9 |

244 | f 0 is a point on the circle and ( P ) is a point in the exterior of the circle. Length of ( boldsymbol{O} boldsymbol{P}=7.5 mathrm{cm} ) and radius of the circle is ( 5.5 mathrm{cm} . ) What will be the length of ( Q P ) if ( Q ) is the centre? A. ( 5.5 mathrm{cm} ) B. ( 13 mathrm{cm} ) ( c .7 .5 mathrm{cm} ) D. ( 13.5 mathrm{cm} ) | 9 |

245 | Find diameter ( & ) circumference with radius ( 7.7 mathrm{cm} ) | 9 |

246 | In the figure ( A O C ) is a diameter of the circle and are ( overline{boldsymbol{A} times boldsymbol{B}}=frac{1}{2} overline{boldsymbol{B} boldsymbol{Y} boldsymbol{C}} ). Find ( angle B O C ) | 9 |

247 | In the given figure, ( A O B ) is a diameter of the circle with center ( boldsymbol{O} ) and ( boldsymbol{A} boldsymbol{C} ) is a tangent to the circle at ( A ). If ( angle B O C= ) ( 130^{circ}, ) then find ( angle A C O ) | 9 |

248 | A circular park of radius ( 20 m ) is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone. A ( .5 sqrt{3} mathrm{cm} ) B. ( 2 sqrt{3} mathrm{cm} ) c. ( sqrt{3} mathrm{cm} ) D. ( 20 sqrt{3} mathrm{cm} ) | 9 |

249 | The center of a circle represented by the equation ( (x-2)^{2}+(y+3)^{2}=100 ) is located in Quadrant ( A ) B. I ( c ) ( D cdot|v| ) | 9 |

250 | ( f(Omega A=5 mathrm{cm}, A B=8 mathrm{cm} text { and } O D ) is perpendicular to ( A B, ) then ( C D ) is equal to: ( A cdot 2 mathrm{cm} ) B. ( 3 mathrm{cm} ) ( c .4 mathrm{cm} ) ( 0.5 mathrm{cm} ) | 9 |

251 | Find the total cost of wooden fencing around a circular garden of diameter ( 28 m . ) If ( 1 m ) of fencing costs 2300 | 9 |

252 | The radius of a circle with centre ( boldsymbol{P} ) is ( 25 mathrm{cm} ) and the length of the chord is 48 ( mathrm{cm} . ) The distance of the chord from centre ( P ) of the circle is ( mathbf{A} cdot 24 mathrm{cm} ) B. ( 5 mathrm{cm} ) ( c .7 mathrm{cm} ) D. ( 12 mathrm{cm} ) | 9 |

253 | Let ( A B C ) be an equilateral triangle inscribed in circle ( 0 . ) M is a point on arc BC. Lines AM, BM and CM are drawn. Then ( overline{boldsymbol{A} boldsymbol{M}} ) is: A. equal to ( overline{B M}+overline{C M} ) B. Less than ( overline{B M}+overline{C M} ) c. greater than ( overline{B M}+overline{C M} ) D. equal, less than, or greater than ( overline{B M}+overline{C M} ) depending upon the pos | 9 |

254 | ( O ) is the centre of the circle. If chord ( A B ) chord ( C D, ) then value of ( x ) is equal to 4.70 3.50 ( c cdot 55 ) ) . 45 | 9 |

255 | In the figure, ( angle A C B=90^{0} ) and radius of big circle ( =2 c m, ) then the radius of small circle is (in ( c m) ) ( A cdot 3-2 sqrt{2} ) B. ( 4-2 sqrt{2} ) c. ( 7-4 sqrt{2} ) D. ( 6-4 sqrt{2} ) | 9 |

256 | 67. The radius of two concentric cir- cles are 9 cm and 15 cm. If the chord of the greater circle be a tangent to the smaller circle, then the length of that chord is (1) 24 cm (2) 12 cm (3) 30 cm (4) 18 cm | 9 |

257 | The length of the chord of the circle ( (x-3)^{2}+(y-5)^{2}=80 ) cut off by the line ( 3 x-4 y-9=0 ) is A . 16 B. 8 ( c cdot sqrt{96} ) ( mathbf{D} cdot 2 sqrt{96} ) | 9 |

258 | In the figure, ‘O’ is the centre of the circle and ( 0 mathrm{M}, ) On are the perpendiculars from the centre to the chords ( P Q ) and ( R S . ) If ( O M=O N ) and ( P Q=6 ) ( mathrm{cm} . ) Find RS | 9 |

259 | Consider a circle of radius ( R ). What is the length of a chord which subtends an angle ( theta ) at the centre? ( ^{mathrm{A}} cdot_{2 R sin frac{theta}{2}} ) B. ( 2 R sin theta ) c. ( _{2 R tan frac{theta}{2}} ) D. ( 2 R tan theta ) | 9 |

260 | 70. AB is a diameter of a circle with centre at O. DC is a chord of it such that DC | AB. If ZBAC = 20°, then 2 ADC is equal to (1) 120 (2) 110° (3) 115 (4) 100° | 9 |

261 | What is the volume in cubic cm of a pyramid whose area of the base is 25 sq cm height ( 9 c m ? ) A ( cdot 75 mathrm{cm}^{3} ) B. ( 70 mathrm{cm}^{3} ) ( mathrm{c} cdot 100 mathrm{cm}^{3} ) D. None of these | 9 |

262 | The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord? | 9 |

263 | Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc. | 10 |

264 | 58. Each of the circles of equal radi with centres A and B pass through the centre of one anoth- er circle they cut at C and D then ZDBC is equal to (1) 60° (2) 100° (3) 120° (4) 140° | 9 |

265 | The length of the shortest chord of the circles ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{2} boldsymbol{g} boldsymbol{x}+boldsymbol{2} boldsymbol{f} boldsymbol{y}+boldsymbol{c}=boldsymbol{0} ) which passes through the point ( (a, b) ) inside the circle | 9 |

266 | Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres. | 9 |

267 | A circle has the equation ( (x+1)^{2}+ ) ( (y-3)^{2}=16 ) What are the coordinates of its center and the length of its radius? A. (-1,3) and 4 B. (1,-3) and 4 c. (-1,3) and 16 D. (1,-3) and 16 | 9 |

268 | Line segment joining the centre to any point on the circle is a radius of the circle. A. True B. False | 9 |

269 | Given inside a circle, whose radius is equal to ( 13 mathrm{cm}, ) is a point ( mathrm{M} ) at a distance ( 5 mathrm{cm} ) from the centre of the circle. A chord ( A B=25 mathrm{cm} ) is drawn through M. The lengths of the segments into which the chord ( A B ) is divided by the point ( M ) in ( C M ) are A. 12,13 в. 14,11 c. 15,10 D. 16, 9 | 9 |

270 | From an external point ( P, ) two tangents PA and ( P B ) are drawn to the circle with center ( 0 . ) Prove that OP is the perpendicular bisector of ( A B ) | 10 |

271 | Find ( Q M ) ( A cdot 13 c m ) 3. ( 12 mathrm{cm} ) ( c .5 mathrm{cm} ) D. ( 8 mathrm{cm} ) | 9 |

272 | Find equation of circle which passes through the point (2,3) and touches the line ( 2 x-3 y-13=0 ) at the point (2,-3) | 10 |

273 | Consider a circle ( x^{2}+y^{2}+a x+b y+ ) ( c=0 ) lying completely in first quadrant. If ( mathrm{m}_{1} ) and ( mathrm{m}_{2} ) are the maximum and minimum values of y/x for all ordered pairs ( (x, y) ) on the circumference of the circle, then the value of ( left(boldsymbol{m}_{1}+boldsymbol{m}_{2}right) ) is A ( cdot frac{a^{2}-4 c}{b^{2}-4 c} ) в. ( frac{2 a b}{b^{2}-4 c} ) c. ( frac{2 a b}{4 c-b^{2}} ) D. ( frac{2 a b}{b^{2}-4 a c} ) | 10 |

274 | The length of tangent drawn from an external point ( P ) to a circle with centre 0, is ( 8 mathrm{cm} ). If the radius of the circle is 6 ( mathrm{cm}, ) then the length of OP (in cm) is : A ( 2 sqrt{7} ) 7 B. ( 4 sqrt{7} ) ( c cdot 10 ) D. 10.5 | 10 |

275 | If ( boldsymbol{A}=(mathbf{5}, mathbf{8}), ) then area of ( triangle boldsymbol{A B D} ) in square units is A ( cdot frac{96 sqrt{5}}{89} ) в. ( frac{960 sqrt{5}}{89} ) c. ( frac{960 sqrt{5}}{sqrt{8} 9} ) D. None of these | 9 |

276 | If two tangents are drawn to a circle circle from an external point, the (i) they subtend equal angles at the centre (ii) they are equally inclined to the segment,joining the centre to that point. | 10 |

277 | Circles with centres ( A, B ) and ( C ) touch each other externally. If ( boldsymbol{A B}= ) ( mathbf{3} c boldsymbol{m}, boldsymbol{B} boldsymbol{C}=mathbf{3} boldsymbol{c m}, boldsymbol{C} boldsymbol{A}=mathbf{4} boldsymbol{c m}, ) then find the radii of each circle | 9 |

278 | A chord of a circle of radius ( 15 mathrm{cm} ) subtends an angle of ( 120^{circ} ) at the centre. Find the area corresponding minor sector of the circle. | 9 |

279 | 69. In a cyclic quadrilateral ABCD, if ZB-ZD = 60° then the measure of the smaller of the two is : (1) 60° (2) 40° (3) 38° (4) 30° | 9 |

280 | The locus of the feet of perpendiculars drawn from the point ( (a, 0) ) on tangents to the circle ( x^{2}+y^{2}=a^{2} ) is A ( cdot a^{2}left(x^{2}+y^{2}+a xright)^{2}=a^{2}left(y^{2}+(x+a)^{2}right) ) B ( cdot a^{2}left(x^{2}+y^{2}-a xright)^{2}=y^{2}+(x-a)^{2} ) C ( cdotleft(x^{2}+y^{2}-a xright)^{2}=a^{2}left(y^{2}+(x-a)^{2}right) ) D cdot a ( ^{2}left[left(x^{2}+y^{2}right)-a^{2} x^{2}right]=left(y^{2}+(x-a)^{2}right) ) | 10 |

281 | A circle passes through (0,0) and (1,0) and touches the circle ( x^{2}+y^{2}=9 ) then the centre of circle is ( ^{mathbf{A}} cdotleft[frac{3}{2}, frac{1}{2}right] ) В. ( left[frac{1}{2}, frac{3}{2}right] ) c. ( left[frac{1}{2}, frac{1}{2}right] ) D. ( left[frac{1}{2}, pm sqrt{2}right] ) | 9 |

282 | f the length of the chord ( Y Z ) is equal to the radius of the circle ( O Y ), find ( angle Y X Z ) A ( cdot 60^{circ} ) B. ( 30^{circ} ) ( c cdot 80^{circ} ) D. 100 | 9 |

283 | Two circles of radii ( 10 mathrm{cm} ) and ( 8 mathrm{cm} ) intersect each other and the length of the common chord is 12 m. Then the distance between their centres is ( mathbf{A} cdot(10+2 sqrt{7}) mathrm{cm} ) B. ( (8+2 sqrt{7}) mathrm{cm} ) ( mathbf{c} cdot(12+2 sqrt{7}) mathrm{cm} ) D ( cdot(6+2 sqrt{7}) ) ст | 9 |

284 | A chord ( A B ) is at a distance of ( 6 mathrm{cm} ) from the centre of a circle whose radius is ( 6 mathrm{cm} ) less than that of the chord ( A B ) Then the length of the chord ( A B ) is ( A cdot 8 mathrm{cm} ) B. ( 32 mathrm{cm} ) c. ( 24 mathrm{cm} ) D. ( 16 mathrm{cm} ) | 9 |

285 | If a chord of a circle ( x^{2}+y^{2}=32 ) makes equal intercepts of length ( l ) on the co-ordinate axes, then ( mathbf{A} cdot ell<8 ) в. ( ell8 ) D. ( ell>16 ) | 9 |

286 | Prove that the line joining the mid- points of two parallel chords of a circle passes through the centre. | 9 |

287 | Draw two tangents from a point ( 5 mathrm{cm} ) away from the centre of a circle of radius ( 3 mathrm{cm} ) | 10 |

288 | Find the value of ( x ) in each of the following diagrams ( (mathbf{i}) ) (ii) | 9 |

289 | If radius of circle is ( 5 mathrm{cm} ) and distance from centre to the point of intersection of 2 tangents in ( 13 mathrm{cm} . ) Find length of tangent. A . ( 11 mathrm{cm} ) B. ( 10 mathrm{cm} ) c. ( 12 mathrm{cm} ) D. ( 13 mathrm{cm} ) | 10 |

290 | Equation of chord ( mathrm{AB} ) of circle ( x^{2}+ ) ( boldsymbol{y}^{2}=boldsymbol{2} ) passing through (2,2) such that ( boldsymbol{P B} / boldsymbol{P A}=mathbf{3}, ) is given by A ( . x=3 y ) В. ( y-2=sqrt{3}(x-2) ) c. ( x=y ) D. none of these | 9 |

291 | If two equal chords of a circle intersect within the circle, prove that the chords and line joining the point of intersection to the centre makes angles which are A. Complementary to each other B. Suplimentary to each other c. Equal to each other D. Not equal to each other | 9 |

292 | An equilateral triangle is inscribed in a circle of radius ( 6 mathrm{cm} . ) Find its side. | 9 |

293 | The radius of the circle ( x^{2}+y^{2}+x+ ) ( c=0 ) passing through the origin is A ( cdot frac{1}{4} ) в. ( frac{1}{2} ) c. 1 D. 2 | 9 |

294 | A rectangle ABCD is inscribed in a circle with centre 0. If AC is the diagonal and ( angle B A C=30^{circ}, ) then radius of the circle will be equal to A ( cdot frac{sqrt{3}}{2} B C ) B. BC ( c cdot sqrt{3} B C ) D. 2BC | 9 |

295 | 65. In the given figure, PAB is a se- cant and PT is a tangent to the circle from P. If PT = 5 cm, PA = 4 cm and AB = x cm, then x is 5 cm P 4 cm A x cm 7В x cm cm (3) 5 cm (43 cm | 10 |

296 | A steel wire, when bent in the form of a square, encloses an area of 121 sq. ( mathrm{cm} ) The same wire is bent in the form of a circle. Find the area of the circle. | 9 |

297 | From the figure, identify a point in the exterior. | 9 |

298 | n Figure, ( P Q ) is a chord of length ( 8 mathrm{cm} ) of a circle of radius ( 5 mathrm{cm} . ) The tangents at ( P ) and ( Q ) intersect at a point ( T . ) The ength of ( T P ) is equal to ( frac{w}{3}, ) then the value of ( a ) is | 10 |

299 | In the given figure, find the value of ( x ) ( A cdot 25 ) 3.30 ( c .35 ) 2.4 | 10 |

300 | Two tangents are drawn to a circle from an external point ( A ), touching the circle at the points ( P ) and ( Q . A ) third tangent intersects segment ( boldsymbol{A P} ) at ( boldsymbol{B} ) and segment ( A Q ) at ( C ) and touches the circle at ( R ) If ( A Q=10 ) units, then the perimeter of ( Delta A B C ) is A . 22.0 в. 20.5 ( c .20 .0 ) D. 40.0 | 10 |

301 | Fill in the blanks The longest chord of a circle is a of the circle. | 9 |

302 | Area of circle in which a chord of length ( 2 sqrt{3} ) units, subtends angle ( 120^{circ} ) at its centre is : A . ( pi ) sq units B. 2 ( pi ) sq units c. ( 4 pi ) sq units D. ( 5 pi ) sq units | 9 |

303 | Let ( S=x^{2}+y^{2}+2 g x+2 f y+c=0 ) be a given circle. Then the locus of the foot of the perpendicular drawn from the origin upon any chords of ( S ) which subtends right angle at the origin is: A ( cdot x^{2}+y^{2}+g x+f y+c / 2=0 ) B . ( x^{2}+y^{2}=g ) c. ( x^{2}+y^{2}=f ) D. ( x^{2}+y^{2}+g=0 ) | 9 |

304 | Two circles of radii ( 20 mathrm{cm} ) and ( 37 mathrm{cm} ) intersect in ( A ) and ( B . ) If ( O_{1} ) and ( O_{2} ) are their centres and ( A B=24 mathrm{cm}, ) then the distance ( O_{1} O_{2} ) is equal to A . ( 44 mathrm{cm} ) B. ( 51 mathrm{cm} ) ( c .40 .5 mathrm{cm} ) D. ( 45 mathrm{cm} ) | 9 |

305 | ( A B ) and ( C D ) are two parallel chords of a circle such that ( A B=10 mathrm{cm}, C D=24 mathrm{cm} ) If the chords are on opposite sides of the centre and distance between them is 17 ( mathrm{cm}, ) the radius of the circle is ( A cdot 10 mathrm{cm} ) B. ( 11 mathrm{cm} ) c. ( 12 mathrm{cm} ) D. ( 13 mathrm{cm} ) | 9 |

306 | In the following figure, ray PA is tangent to the circle at ( A ) and ( P B C ) is a secant. If ( A P=15, B P=10, ) then find ( B C ) | 10 |

307 | If two equal chords of a circle intersect each other, then prove that the segments of one chord are equal to corresponding segment of the other chord. | 9 |

308 | In the figure on your right, 0 is the centre of the circle State Which of the line segment are chords? | 9 |

309 | ( mathrm{M} ) and ( mathrm{N} ) are the mid-points of two equal chords ( A B ) and ( C D ) respectively of ( a ) circle with centre ( 0 . ) Prove that ( angle A M N=angle C N M ) | 9 |

310 | ( O ) is the centre of the circle having radius ( 5 mathrm{cm} . O M ) is a ( perp ) on chord ( A B ). If ( O M=4 mathrm{cm}, ) then the length of the chord ( A B ) is equal to ( A cdot 5 mathrm{cm} ) B. ( 6 mathrm{cm} ) ( c cdot 8 mathrm{cm} ) D. ( 10 mathrm{cm} ) | 9 |

311 | The longest chord of a circle is a ( ldots ). Of the circle. A. Diameter B. Lies on upper part of centre c. Lies on lower part of centre D. None of these | 9 |

312 | Given point are ( boldsymbol{P}=(mathbf{1},-mathbf{2}), boldsymbol{Q}=(mathbf{7}, mathbf{6}) ) is the origin. The length of the common chord of the circles with ( mathrm{OP} ) and ( mathrm{OQ} ) as diameters is A . 1 B. 2 ( c cdot 4 ) D. 6 | 9 |

313 | Two circles of radii ( 10 mathrm{cm} ) and ( 8 mathrm{cm} ) intersects each other and the length of the common chord is ( 12 mathrm{cm} ), find the distance between their centers. A. ( 2 mathrm{cm} ) в. ( (8+2 sqrt{7}) ) ст ( c .8 mathrm{cm} ) D. ( 2 sqrt{7} ) cm | 9 |

314 | A circle has two equal chords ( $ $ P Q $ $ ) and ( $ $ P R $ $ ) diameter ( $ $ P D $ $ ) cuts ( $ $ Q R $ $ ) in ( $ $ E ) ( $ $ . ) If ( P R=12 c m ) and ( P E=8 c m, ) then the length of ( $ $ ) PD ( $ $ ) is ( ? ) ( mathbf{A} cdot 25 mathrm{cm} ) B . ( 22 mathrm{cm} ) c. ( 20 mathrm{cm} ) D. ( 18 mathrm{cm} ) | 9 |

315 | Prove that if chords of congruent circles subtend equal angles their centres, then the chords are equal. | 9 |

316 | Find the equation of a circle with centre (2,2) and passes through the point (4,5) | 9 |

317 | ( angle A O B ) is ( mathbf{A} cdot 54^{circ} ) B. ( 72^{circ} ) ( c cdot 90^{circ} ) ( D cdot 108^{circ} ) | 10 |

318 | 69. PO and OR are two chords of a circle and they are equally in- clined to the diameter drawn through Q. What is the relation between PO and QR? (1) PO 1 OR (2) PO > QR (3) PQ < OR (4) PO = QR | 10 |

319 | ( A B ) and ( C D ) are two parallel chords of a circle such that ( A B=10 mathrm{cm} ) and ( mathrm{CD}=24 ) ( mathrm{cm}, ) If the chords are on the opposite sides of the centre and the distance between them is ( 17 mathrm{cm} ), the radius of the circle is: A . ( 14 mathrm{cm} ) B. 10 ( mathrm{cm} ) ( c cdot 13 mathrm{cm} ) D. ( 15 mathrm{cm} ) | 9 |

320 | The line drawn from center of circle to bisect a chord is perpendicular to the chord. Is this true? If true enter 1 else 0 . | 9 |

321 | 59. Inscribed ZACB intercepts AB of circle with centre 0. If the bisec- tor of ZACB meets arc AB in M then : (1) m AM > m MB (2) m AM<m MB (3) m AM = m MB (4) None of these | 9 |

322 | If the lines ( 3 x-4 y+4=0 ) and ( 6 x- ) ( 8 y-7=0 ) are tangents to a circle, then find the radius of the circle. ( A cdot 3 / 4 ) в. ( 4 / 3 ) ( c cdot 1 / 4 ) D. ( 7 / 4 ) | 9 |

323 | In the adjacent figure, ( A B ) is a chord of circle with centre ( 0 . ) CD is the diameter perpendicular to AB. Show that ( A D=B D ) | 9 |

324 | Prove that the line joining a mid point of a chord to the centre of circle is perpendicular to it | 9 |

325 | Hilswer untulury quesuuris Na>tu on the following. ( C_{1} ) and ( C_{2} ) are two circles and points ( boldsymbol{P}_{1}, boldsymbol{P}_{2}, boldsymbol{P}_{3}, boldsymbol{P}_{4}, boldsymbol{P}_{5} ) are noted. From which point tangent is possible to ( C_{2} ) but not ( C_{1} ) ( A cdot P_{2} ) в. ( P_{3} ) ( c cdot P_{4} ) D. ( P_{5} ) | 10 |

326 | Find the centre and radius of the circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+2 boldsymbol{a} boldsymbol{x}-boldsymbol{2} boldsymbol{b} boldsymbol{y}+boldsymbol{b}^{2}=mathbf{0} ) | 9 |

327 | Prove that the lengths of the tangents drawn from an external point to a circle are equal. | 10 |

328 | 69. If O be the circumcentre of a tri- angle PQR and Z QOR= 110°, 2 OPR = 25°, then the measure of PRO is (1) 65° (2) 50° (3) 55° (4) 60° | 9 |

329 | A chord of a circle is ( 12 mathrm{cm} ) which is at a distance of ( 8 mathrm{cm} ) from center. Find the length of the chord of the same circle which is at a distance of ( 6 mathrm{cm} ) from the centre ( A cdot 20 mathrm{cm} ) B. ( 24 mathrm{cm} ) ( c .16 mathrm{cm} ) D. ( mathrm{cm} ) | 9 |

330 | Draw a circle and mark a diameter. | 9 |

331 | The chord of a ( odot(0,5) ) touches ( odot(0,3) ) The length of the chord is ( mathbf{A} cdot mathbf{8} ) B. 6 ( c cdot 7 ) D. | 9 |

332 | ( P ) is a point on the common chord ( R S ) produced by two intersecting circles. ( A B ) and ( C D ) are the chords of the circles,they meet at ( P ) produced.Prove that ( boldsymbol{P A} times boldsymbol{P B}=boldsymbol{P C} times boldsymbol{P D} ) | 9 |

333 | In the given figure, ( P A ) and ( P B ) are two tangents drawn from an external point ( P ) to a circle with centre ( O ). Prove that OP is the right bisector of line segment ( A B ) | 10 |

334 | 70. ABCD is a cyclic quadrilateral. The side AB is extended to E in such a way that BE=BC. If ZADC = 70°, ZBAD = 95°, then ZDCE is equal to (1) 140° (2) 120° (3) 165 (4) 110° | 9 |

335 | Find the value of ( x ) A ( cdot 50^{circ} ) B ( .60^{circ} ) ( c cdot 70^{circ} ) D. ( 80^{circ} ) | 10 |

336 | Find the angle marked ( a ) 4.77 8. 36 ( c cdot 41^{circ} ) ( mathbf{D} cdot 13^{circ} ) | 10 |

337 | The inner circumference of a circular track is ( 24 pi mathrm{m} ). The track is ( 2 mathrm{m} ) wide from everywhere. The quantity of wire required to surround the path completely is A. ( 80 mathrm{m} ) B. ( 81 mathrm{m} ) c. ( 82 mathrm{m} ) D. 88m | 9 |

338 | In the following figure, ( Delta A B C ) is an isosceles triangles with perimeter ( 40 mathrm{cm} . ) The base ( A C ) is of length ( 10 mathrm{cm} ) Side ( A B ) and side ( B C ) are congruent. ( A ) circle touches the three sides as shown in the figure below. Find the length of the tangent segment from point ( B ) to the circle. | 10 |

339 | n given figure ( C ) is centre of circle. ( A O ) and ( B O ) are tangents to circle. ( C M perp ) ( A B ) ( mathbf{f} boldsymbol{A} boldsymbol{C}=boldsymbol{A} boldsymbol{B}=mathbf{6} boldsymbol{c m}, ) then find ( boldsymbol{A} boldsymbol{M} ) ( A cdot 3 c m ) 8. 4 ст ( c .5 mathrm{cm} ) D. ( 6 mathrm{cm} ) | 9 |

340 | Tangents are drawn from (4,4) to the ( operatorname{circle} x^{2}+y^{2}-2 x-2 y-7=0 ) meet the circle at ( A ) and ( B ). The length of the chord ( A B ) is A ( cdot 2 sqrt{3} ) 3 B. ( 3 sqrt{2} ) c. ( 2 sqrt{6} ) D. ( 6 sqrt{2} ) | 9 |

341 | Name the following part from the adjacent figure where ‘O’ is the center of the circle. ( boldsymbol{A O} ) | 9 |

342 | n the following figure, the line ABCD is perpendicular to PQ ; where P and Q are the centres of the circles. Show that: ( A B=C D ) ii) ( A C=B D ) | 9 |

343 | Prove that if chords of congruent circles subtend equal angles at their centre, then the chords are equal. | 9 |

344 | In two concentric circle, prove that all chords of the outer circle which touch the inner are of equal length. | 9 |

345 | Find the equation of the circle whose center lies on the positive direction of ( y ) axis at a distance 6 from the origin and whose radius is 4 | 9 |

346 | Prove that the angle in a semicircle is a right angle. | 9 |

347 | 67. O and C are respectively the or- thocentre and circumcentre of an acute-angled triangle PQR. The points P and O are joined and produced to meet the side QR at S. If ZPQS = 60° and ZQCR = 130°, then ZRPS = (1) 30° (2) 35° (3) 100 (4) 60° | 9 |

348 | If ( operatorname{lines} x-2 y+3=0,3 x+k y+7= ) 0 cut the coordinate axes in concyclic points, then ( k=? ) ( mathbf{A} cdot 3 / 2 ) B. ( 1 / 2 ) c. ( -3 / 2 ) D. -4 | 9 |

349 | In the figure ( P Q ) is tangent to the circle at ( p t ) Find the radius, if ( P Q=8 mathrm{cm} ) and ( boldsymbol{O} boldsymbol{R}=mathbf{1 0} boldsymbol{c m} ) | 10 |

350 | Two parallel chords are drawn on the same side of the centre of a circle of radius 20. It is found that they subtend ( 60^{0} ) and ( 120^{0} ) angles at the centre of the circle. Then the perpendicular distance between the chords is: ( mathbf{A} cdot 5(sqrt{3}-1) ) B . ( 10(sqrt{3}-1) ) c. ( 10(sqrt{2}-1) ) () ( 5(sqrt{2}-1-1) ) D. ( 5(sqrt{3}+1) ) | 9 |

351 | A line segment whose end points lie on the circle is called ( ldots ldots . . . . . . . ) to the circle. A. Chord B. tangent c. Radius D. Diameter | 9 |

352 | In a diagram ( boldsymbol{O} ) is the centre of circle. Calculate the value ( a ) A . 43 B. 53 ( c cdot 63 ) D. 33 | 9 |

353 | In the figure, ‘O’ is the centre of the circle. ( O M=3 mathrm{cm} ) and ( mathrm{AB}=8 mathrm{cm} ). Find the radius of the circle. A ( .5 mathrm{cm} ) B. 4 cm c. ( 15 mathrm{cm} ) D. ( 8 c m ) | 9 |

354 | n figure if PQR is tangent to circle at ( mathrm{Q} ) whose centre is ( 0 . A B ) is a chord parallel to PR and ( angle B Q R=70^{circ} ) then ( angle A Q B ) is equal to ( mathbf{A} cdot 20 ) B. 40 ( c .35 ) D. 45 | 10 |

355 | Establish the formula for area and circumference of circle. | 9 |

356 | If ( L equiv 2 x+y-6=0, ) then the locus of circumcentre of ( triangle P Q R ) is A. ( 2 x-y=4 ) в. ( 2 x+y=3 ) c. ( x-2 y=4 ) D. ( x+2 y=3 ) | 10 |

357 | A tangent ( mathrm{PQ} ) at a point ( mathrm{P} ) of a circle of radius ( 5 mathrm{cm} ) meets a line through the centre 0 at a point ( Q, ) so that ( O Q=12 ) cm. Length of PQ is : A. ( sqrt{112} mathrm{cm} ) B . ( sqrt{113} mathrm{cm} ) c. ( sqrt{85} mathrm{cm} ) D. ( sqrt{119} mathrm{cm} ) | 10 |

358 | 55. AB is the diameter of circle and AC is its one chord. The tangent at C intersect the produced di- ameter AB at D. Given that AB = 10 cm, AC = 8 cm ZBAC = 30° then BD will be equal to (1) 6 cm (2) 8 cm (3) 10 cm (4) 4 cm | 9 |

359 | If figure ( C E ) and ( D E ) are equal chords of a circle with centre ( O ). If ( angle A O B= ) ( 90^{circ}, ) find ratio of the area of ( triangle C E D ) and ( triangle A O B ) | 9 |

360 | ( boldsymbol{S} boldsymbol{R}=? ) ( 4 . overline{P Q} ) 3. ( overline{P Q} ) : Q ( . overline{S R} ) | 9 |

361 | The length of a tangent from a point ( boldsymbol{A} ) at distance ( 5 mathrm{cm} ) from the centre of the circle is ( 4 mathrm{cm} ). Find the radius of the circle. | 10 |

362 | A chord of a circle divides the circular region in two parts the region which contains the centre is known as A. minor Arc B. major Arc c. minor Segment D. major Segment | 9 |

363 | Draw a pair of tangents to a circle of radius ( 3 mathrm{cm} ) which are inclined to each other at an angle of ( 45^{circ} ) | 10 |

364 | Coordinates of the centre of the circle which bisects the circumferences of the circles ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=mathbf{1}: boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{2} boldsymbol{x}- ) ( mathbf{3}=mathbf{0} ) and ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+mathbf{2} boldsymbol{y}-mathbf{3}=mathbf{0} ) is A ( cdot(-3,-3) ) B. (3,3) c. (2,2) D. (-2,-2) | 9 |

365 | Two circles with centres ( A ) and ( B ) of radii ( 3 mathrm{cm} ) and ( 4 mathrm{cm}, ) respectively intersect at two points ( C ) and ( D ) such that ( A C ) and ( B C ) are tangents to the two circles. Find the 10 times length of the common chord ( C D ) A .48 B. 58 ( c cdot 56 ) D. 54 | 9 |

366 | What is a line passing through two points on a circle called? A. secant B. Digonal c. Radius D. tangent | 10 |

367 | The common chord of the circles ( x^{2}+ ) ( boldsymbol{y}^{2}-mathbf{4} boldsymbol{x}-mathbf{4} boldsymbol{y}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}^{2}+mathbf{2} boldsymbol{y}^{2}=mathbf{3} mathbf{2} ) subtends at the origin an angle equal to A ( cdot frac{pi}{3} ) B. c. D. | 9 |

368 | In the above figure, 0 is the centre of the circle. The angle ( C B D ) is equal to ( A cdot 25 ) B. 50 ( c cdot 40^{circ} ) D. 130 | 9 |

369 | Tangents are drawn from the point ( (a, a) ) to the circle ( x^{2}+y^{2}-2 x-2 y- ) ( 6=0 . ) If the angle between the tangents lies in the range ( left(frac{pi}{3}, piright), ) then the exhaustive range of values of ( a ) is B. (-5,-3)( cup(3,5) ) c. ( (-infty, 2 sqrt{2}) cup(2 sqrt{2}, infty) ) D. (-3,-1)( cup(3,5) ) | 10 |

370 | If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord. | 9 |

371 | 65. The chord of a circle is equal to its radius. The angle subtended by this chord at the minor arc of the circle is (1) 75° (2) 60° (3) 150 (4) 120° | 9 |

372 | n fig, ( O ) is the centre of a circle ( A B= ) ( mathbf{1 6} mathrm{cm}, boldsymbol{C D}=mathbf{1 4} mathrm{cm}, operatorname{seg} boldsymbol{O} boldsymbol{M} perp mathbf{s e g} ) ( A B, operatorname{seg} O N perp operatorname{seg} C D . ) If ( O M=6 mathrm{cm} ) then length of ( operatorname{seg} O N ) is ( sqrt{m} mathrm{cm} . ) So, ( m ) is A ( . m=149 c m^{2} ) В ( cdot m=51 c m^{2} ) ( mathrm{c} cdot m=51 mathrm{cm} ) D. ( m=149 mathrm{cm} ) | 9 |

373 | OA.OB are the radii of a circle with ( O ) as centre, the angle ( A O B=120^{circ} ) Tangents at ( A ) and ( B ) are drawn to meet | 10 |

374 | In a circle of radius ( 25 mathrm{cm} ) two parallel chords of the length ( 14 mathrm{cm} ) and ( 48 mathrm{cm} ) respectively, are drawn on the same side of the centre. The distance between them is A . ( 14 mathrm{cm} ) B. ( 24 mathrm{cm} ) ( c cdot 17 mathrm{cm} ) D. ( 31 mathrm{cm} ) | 9 |

375 | f two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords. | 9 |

376 | The moon’s distance from the earth is ( 360000 mathrm{km} ) and its diameter subtends an angle of ( 42^{prime} ) at the eye of the observer. The diameter of the moon is? A. ( 4400 mathrm{km} ) B . ( 1000 mathrm{km} ) ( mathbf{c} .3600 mathrm{km} ) D. ( 8800 mathrm{km} ) | 9 |

377 | Find the radius of a circle whose diameter has endpoints (-3,-2) and ( (7, ) 8) A. 5 B. ( 5 sqrt{2} ) ( c cdot(2,3) ) D. ( sqrt{52} ) E. none of these | 9 |

378 | Consider the following diagram where ( A B ) and ( C D ) are congruent arcs and chords. The measure of ( angle A O B=50^{circ} ) Then the value of ( angle C O D=? ) A ( .45^{circ} ) В. ( 50^{circ} ) ( c cdot 56^{circ} ) D. ( 90^{circ} ) | 9 |

379 | Perimeter of a circle is called its A . circumference B. area c. diameter D. none of these | 9 |

380 | n the figure if ( angle B D C=30^{circ}, angle ) ( C B A=110^{circ}, ) then find ( angle B C A ) 4.20 3.40 235 ; 0 | 9 |

381 | ( A, B, C ) are three points on a circle such that ( A B ) is the chord and ( C P ) is the perpendicular to ( O P, ) where ( O ) is the centre and ( P ) is any point on ( A B . ) The radius ( r ) of the circle is given by A ( cdot r^{2}=O P^{2}+A P times C P ) B . ( r^{2}=O P^{2}+A P times P B ) c. ( r^{2}=O P^{2}+P B times P C ) D . ( r^{2}=O P^{2}+P B^{2} ) | 9 |

382 | Calculate the length of a chord which is at a distance of ( 12 mathrm{cm} ) from the centre of a circle of radius ( 13 mathrm{cm} ) | 9 |

383 | In the given figure, ( P Q mathrm{cm}, M ) is the mid-point of ( boldsymbol{Q} boldsymbol{R} ) ? Also, ( M N perp P R, Q S=7 mathrm{cm} ) and ( T R= ) ( 21 c m, ) then ( M N=? ) ( mathbf{A} cdot 14 mathrm{cm} ) ( mathbf{B} cdot 12.5 mathrm{cm} ) c. ( 31 mathrm{cm} ) D. 25 cm | 10 |

384 | Distance of chord ( A B ) from the centre of a circle is ( 8 mathrm{cm} ). Length of the chord ( A B ) is ( 12 mathrm{cm} . ) Find the diameter of the circle | 9 |

385 | 52. Two circles of diameters 10 cm and 6 cm have the same cen- tre. A chord of the larger circle is a tangent of the smaller one. The length of the chord is (1) 4 cm. (2) 8 cm. (3) 6 cm. (4) 10 cm. | 9 |

386 | If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of other chord | 9 |

387 | 71. The tangents drawn at P and Q on the circumference of a circle intersect at A. If Z PAQ = 68°, then the measure of the Z APO (1) 56° (3) 28° (2) 680 (4) 34° | 10 |

388 | The length of the chord of a circle is ( 8 mathrm{cm} ) and perpendicular distance between centre and the chord is ( 3 mathrm{cm} ). Then the radius of the circle is equal to? A. ( 4 mathrm{cm} ) B. ( 5 mathrm{cm} ) ( mathrm{c} cdot 6 mathrm{cm} ) D. ( 8 mathrm{cm} ) | 9 |

389 | The angle between the two tangents from the origin to the circle ( (x-7)^{2}+ ) ( (y+1)^{2}=25 ) equals- ( ^{A} cdot frac{pi}{2} ) в. ( c cdot frac{pi}{4} ) D. None of these. | 10 |

390 | In the given figure given below ( P Q ) is a diameter chord ( S R ) is parallel to ( P Q ) Given ( angle P Q R=58^{circ}, ) calculate ( angle R P Q ) ( A cdot 30^{circ} ) B. 32 ( c cdot 34 ) ( D .36 ) | 9 |

391 | Draw any circle and mark a sector | 9 |

392 | Prove that the line joining the mid- points of two equal chords of a circle subsent equal angles with the chord. | 9 |

393 | 71. ord PQ is 2 is the per PO at M In a given circle, the chord Po of length 18 cm. AB is the pendicular bisector of PQ af If MB = 3 cm, then the length AB is LA un of 79 (1) 27 cm. (3) 28 cm. (2) 30 cm. (4) 25 cm. | 9 |

394 | Tangents PA and PB drawn to ( x^{2}+y^{2}= ) 9 from any arbitrary point ‘P’ on the line ( x+y=25 . ) Locus of midpoint of chord ( A B ) is A ( cdot 25left(x^{2}+y^{2}right)=9(x+y) ) B . ( 25left(x^{2}+y^{2}right)=3(x+y) ) C. ( 5left(x^{2}+y^{2}right)=3(x+y) ) D. None of these | 10 |

395 | In the figure given, ( O ) is the centre of the circle. ( A B ) and ( C D ) are two chords of the circle. ( O M ) is perpendicular to ( A B ) and ( O N ) is perpendicular to ( C D . A B= ) ( mathbf{2 4} c boldsymbol{m}, boldsymbol{O} boldsymbol{M}=mathbf{5} boldsymbol{c m}, boldsymbol{O} boldsymbol{N}=mathbf{1 2} c boldsymbol{m} . ) Finc the Length of chord ( C D ) | 9 |

396 | Find the centre and radius of the circle ( x^{2}+y^{2}+6 x+8 y-96=0 ) | 9 |

397 | A chord of length ( 30 mathrm{cm} ) is drawn at a distance of ( 8 mathrm{cm} ) from the centre of a circle. The radius of the circle (in cm.) is ( mathbf{A} cdot 15 mathrm{cm} ) B. ( 21 mathrm{cm} ) c. ( 18 mathrm{cm} ) D. ( 17 mathrm{cm} ) | 9 |

398 | If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord | 9 |

399 | Prove that the length of the common chord of the two circles whose equations are ( (x-a)^{2}+(y-b)^{2}=c^{2} ) and ( (x-b)^{2}+(y-a)^{2}=c^{2} ) is ( sqrt{4 c^{2}-2(a-b)^{2}} ) Hence find the condition that the two circles may touch. | 9 |

400 | In the figure, the chord BD is perpendicular to the diameter AC. Find the measures of the following angles. a. ( angle B A C ) b. ( angle B C D ) c. ( angle boldsymbol{A} boldsymbol{D} boldsymbol{C} ) ( mathrm{d} . angle C D M ) e. ( angle B A P ) | 9 |

401 | If two chords of lengths ( 2 a ) each, of a circle of radius ( R, ) intersect each other at right angles then the distance of their point of intersection from the centre of the circle is A ( cdot 2 sqrt{R^{2}-a^{2}} ) the ( sqrt{R^{2}-a^{2}} ) B . ( sqrt{2left(R^{2}-a^{2}right)} ) c. ( 4 sqrt{left(R^{2}-a^{2}right)} ) D ( cdot 2left(R^{2}-a^{2}right) ) | 9 |

402 | If tangents ( boldsymbol{T} boldsymbol{A} ) and ( boldsymbol{T} boldsymbol{B} ) from a point ( boldsymbol{T} ) to a circle with centre ( O ) are inclined to each other at an angle of ( 70^{circ}, ) then find ( angle A O B ) (in degrees) | 10 |

403 | ( O ) is centre of the circle. Find the length of radius, if the chord of length ( 24 mathrm{cm} ) is at a distance of ( 9 mathrm{cm} ) from the centre of the circle. | 9 |

404 | In the figure given above, ( A D ) is a straight line, ( O P ) perpendicular to ( A D ) and 0 is the centre of both circles. If ( boldsymbol{O A}=mathbf{2 0} boldsymbol{c m}, boldsymbol{O B}=mathbf{1 5} boldsymbol{c m} ) and ( boldsymbol{O P}= ) ( 12 mathrm{cm} . ) what is ( A B ) equal to? ( A cdot 7 mathrm{cm} ) ( 3.8 mathrm{cm} ) ( c .10 mathrm{cm} ) ( 0.12 mathrm{cm} ) | 9 |

405 | Prove that out of all the chords which passing through any point circle, that chord will be smallest which is perpendicular on diameter which passes through that point. | 9 |

406 | The distance from the centre to the circumference. A. Sector B. Segment c. Diameters D. Radius | 9 |

407 | ( A B ) and ( C D ) are two parallel chords of a circle such that ( A B=10 mathrm{cm} ) and ( C D=24 mathrm{cm} . ) If the chords are on opposite sides of the centre and the distance between them is ( 17 mathrm{cm} . ) The radius of the circle is ( mathbf{A} cdot 26 mathrm{cm} ) B. ( 39 mathrm{cm} ) c. ( 6.5 mathrm{cm} ) D. ( 13 mathrm{cm} ) | 9 |

408 | 67. If a chord of length 16 cm is at a distance of 15 cm from the cen- tre of the circle, then the length of the chord of the same circle which is at a distance of 8 cm from the centre is equal to (1) 10 cm (2) 20 cm (3) 30 cm (4) 40 cm | 9 |

409 | Two parallel chords in a circle are ( 10 mathrm{cm} ) and ( 24 mathrm{cm} ) long. If the radius of the circle is ( 13 mathrm{cm} ), find the distance between the chords if thay lie on the opposite sides of the center | 9 |

410 | Two equal circles in the same plane can have at the most the following numbers of common tangents ( A cdot 3 ) B . 2 ( c cdot 4 ) D. | 10 |

411 | Which is a secant? ( A cdot ) ми B. on ( c . P Q ) D. None | 10 |

412 | 70. Two chords AB and CD of cri- cle whose centre is O, meet at the point P and 2 AOC = 50° BOD = 40°. Then the value of BPD is (1) 60° (2) 40° (3) 45° (4) 75° | 9 |

413 | Find the area of the sector of a circle whose radius is ( 14 mathrm{cm} ) and angle of sector is ( 45^{circ} ) | 9 |

414 | The tangents drawn from the origin to the circle ( x^{2}+y^{2}+2 g x+2 f y+f^{2}= ) 0 are perpendicular, if A. ( g=f ) в. ( g=2 f ) c. ( 2 g=f ) D. ( 3 g=f ) | 10 |

415 | Find the coordinates of a point ( A, ) where ( A B ) is the diameter of a circle whose centre is (2,-3) and ( B ) is (1,4) | 9 |

416 | If ( alpha ) is the angle subtended at ( Pleft(x_{1}, y_{1}right) ) by the circle ( S=x^{2}+y^{2}+2 g x+ ) ( 2 f y+c=0, ) then This question has multiple correct options A ( cdot cot alpha=frac{sqrt{S_{1}}}{sqrt{g^{2}+f^{2}-c}} ) B. ( cot alpha / 2=frac{sqrt{S_{1}}}{sqrt{g^{2}+f^{2}-c}} ) ( ^{mathrm{c}} tan alpha=frac{2 sqrt{g^{2}+f^{2}-c}}{sqrt{S_{1}^{1}}} ) D. ( quad alpha=2 tan ^{-1}left(frac{sqrt{g^{2}+f^{2}-c}}{sqrt{S_{1}}}right) ) | 10 |

417 | What are the coordinates of the center of this circle? ( boldsymbol{x}^{2}+(boldsymbol{y}+mathbf{7})^{2}=mathbf{1 1} ) ( A cdot(7,7) ) B. (0,7) c. (-7,-7) D. (0,-7) | 9 |

418 | If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord. | 9 |

419 | In the given figure, 0 is the centre of the circle and ( angle A B C=36^{circ} . ) The measure of ( angle A O C ) is : ( A cdot 36 ) B. 72 ( c cdot 144 ) ( D cdot 18 ) | 9 |

420 | 70. Two circles of radii Rand r touch each other externally and PQ is the direct common tangent, Then PO2 is equal to: (1) R-T (2) R+T (3) 2R (4) 4R | 10 |

421 | Find ( boldsymbol{P} boldsymbol{M} ) ( mathbf{A} cdot 3 c m ) ( mathbf{B} cdot 4 c m ) ( mathbf{c} cdot 5 c m ) D. ( 8 mathrm{cm} ) | 9 |

422 | ( O ) is the centre of the circle with radius ( 5 mathrm{cm} . ) Chords ( A B ) and ( C D ) are parallel. ( A B=6 mathrm{cm} ) and ( C D=8 mathrm{cm} . ) If ( P Q ) is distance between ( A B ) and ( C D ), then the length of ( boldsymbol{P Q} ) is ( mathbf{A} cdot 10 mathrm{cm} ) B. ( 8 mathrm{cm} ) ( c cdot 7 mathrm{cm} ) D. ( 7 sqrt{2} mathrm{cm} ) | 9 |

423 | The length of a minor arc is ( frac{2}{9} ) of the circumference of the circle. Write the measure of the angle subtended by the arc at the centre of the circle. | 9 |

424 | Given ( B D=12 ) and ( A C=3 ) in the circle with center ( A ). Find the radius. A . 3 B. ( 3 sqrt{5} ) ( c cdot 4 ) D. ( 4 sqrt{5} ) | 9 |

425 | The common point of the tangent and circle is called A. Intersecting points B. Secant c. Point of contact D. None | 10 |

426 | The radius of a circle with centre 0 is 13 ( mathrm{cm} . ) The distance of a chord from the centre is ( 5 mathrm{cm} . ) Find the length of the chord. ( mathbf{A} cdot 24 mathrm{cm} ) B. ( 12 mathrm{cm} ) ( mathrm{c} cdot 13 mathrm{cm} ) D. ( 26 mathrm{cm} ) | 9 |

427 | n the figure, ( P Q=R S ) and ( angle O R S= ) ( 48^{circ} ) Find ( angle O P Q ) and ( angle R O S ) | 9 |

428 | The length of chord of circle with radius 10cm drawn at a distance of 8cm ( A cdot 12 mathrm{cm} ) B. ( 10 mathrm{cm} ) ( c cdot 14 c m ) D. 30cm | 9 |

429 | In the given figure 0 is the centre of the circle ( 0 B=5 c m, ) Distance from 0 to Chord ( A B ) is ( 3 c m ).Find the length of ( A B ) | 9 |

430 | From the following figure find value of ( boldsymbol{X} boldsymbol{Y} ) if ( angle boldsymbol{A} boldsymbol{O} boldsymbol{B}=angle boldsymbol{X} boldsymbol{O} boldsymbol{Y} ) 3. 11 ( r ) 2 | 9 |

431 | 66. Each of the circles of equal radit with centres A and B pass through the centre of one anoth- er circle they cut at C and D then ZDBC is equal to (1) 60° (2) 100° (3) 120 (4) 140° | 9 |

432 | Find the length of longest chord of the circle if radius is ( 2.9 mathrm{cm} ) in ( mathrm{cm} ) | 9 |

433 | At one end ( A ) of a diameter ( A B ) of a circle of radius ( 5 mathrm{cm}, ) tangent ( mathrm{XAY} ) is drawn to the circle. Find the length of the chord CD parallel to XY and at a distance ( 8 mathrm{cm} ) from A. | 9 |

434 | Two equal chords of a circle intersect within the circle Then the corresponding segments of the chords are A . not always equal B. not equal c. not related anyway D. equal | 9 |

435 | In the diagram, 0 is the centre of the circle. The angles CBD is equal to ( A cdot 120 ) B. ( 55^{circ} ) ( c cdot 65 ) D. 75 | 9 |

436 | ( A B ) is a chord of a circle with center 0 The tangent at B cuts AO produced at T if ( angle B A T=25^{circ} ) Then the value of ( angle B T A ) is A . ( 30^{circ} ) B. ( 60^{circ} ) ( c cdot 25 ) D. ( 40^{circ} ) | 9 |

437 | The lengths of the two tangents from an external point to a circle are A. equal B. different c. both A and B D. none of the above | 10 |

438 | In the given figure, ( boldsymbol{T} boldsymbol{T}^{prime} ) is the tangent line. Which one of the following relationship is true? ( mathbf{A} cdot x+y=2 z ) B . ( x+y=z ) c. ( z-3 x=y ) D. ( z-2 x=y ) | 10 |

439 | The value of ( c, ) for which the line ( y= ) ( 2 x+c ) is a tangent to the circle ( x^{2}+ ) ( boldsymbol{y}^{2}=mathbf{1 6}, ) is A . ( -16 sqrt{5} ) B. 20 c. ( 4 sqrt{5} ) D. ( 16 sqrt{5} ) | 10 |

440 | The equation to the sides ( A B, B C, C A ) of a ( triangle operatorname{are} boldsymbol{x}+boldsymbol{y}=mathbf{1} ; mathbf{4} boldsymbol{x}-boldsymbol{y}+mathbf{4}= ) and ( 2 x+3 y=6 . ) Circle are drawn on ( A B, B C, C A ) as diameter. The point of concurrence of the common chord is A. centroid of the triangle B. orthocenter c. circumcenter D. incenter | 9 |

441 | In the given figure, 0 is the centre of the circle. If ( angle A O D=140^{circ} ) and ( angle C A B= ) ( 50^{circ}, ) Then (i) ( angle boldsymbol{E} boldsymbol{D} boldsymbol{B} ) (ii) ( angle boldsymbol{E} boldsymbol{B} boldsymbol{D} ) are respectively ( begin{array}{lll}A & -70^{circ} & & 50end{array} ) ( begin{array}{lll}text { B } cdot 50^{circ} & 110^{circ}end{array} ) ( begin{array}{lll}c cdot 30^{circ} & & 70^{circ}end{array} ) ( begin{array}{lll}text { D. } 120^{circ} & text { & } 130^{circ}end{array} ) | 9 |

442 | Length of the common chord of the ( operatorname{circles}(x-1)^{2}+(y+1)^{2}= ) ( c^{2} ) and ( (x+1)^{2}+(y-1)^{2}=c^{2} ) is A ( cdot frac{1}{2} sqrt{c^{2}-2} ) B. ( sqrt{c^{2}-2} ) c. ( 2 sqrt{c^{2}-2} ) D. ( (c+2) ) | 9 |

443 | A line touches a circle of radius ( 4 mathrm{cm} ) Another line is drawn which is tangent to the circle. If the two lines are parallel then distance between them is A ( .4 mathrm{cm} ) в. 6 ст ( c .7 c m ) D. ( 8 mathrm{cm} ) | 10 |

444 | If the line ( y-m x+m-1=0 ) cuts the ( operatorname{circle} x^{2}+y^{2}-4 x-4 y+4=0 ) at two real points, then ( m ) belongs to A . [1,1] B . [-2,2] ( c cdot(-infty, infty) ) D. [-4,4] | 9 |

445 | The equation of the diameter of circle ( x^{2}+y^{2}+2 x-4 y-11=0 ) which bisects the chords intercepted on the line ( 2 x-y+3=0 ) is A. ( x+y-7=0 ) В. ( 2 x-y-5=0 ) c. ( x+2 y-3=0 ) D. None of these | 10 |

446 | If ( omega ) is a cube root of unity, then ( (3+ ) ( left.mathbf{5} boldsymbol{omega}+mathbf{3} boldsymbol{omega}^{2}right)^{2}+left(mathbf{3}+mathbf{3} boldsymbol{omega}+mathbf{5} boldsymbol{omega}^{2}right)^{2} ) is equal to A . 4 B. ( c cdot-4 ) D. None of these | 9 |

447 | 8. In a triangle ABC, let C = Ifr is the inradius and R is Toumradius of the triangle ABC, then 2 (r+R) equals [2005] (a) b+c (b) a + b (c) a+b+c (d) cta atte | 9 |

448 | In which circles, angles at the centers make a equal chords? A. concentric circles B. eccentric circles c. tangential circles D. equal circles | 9 |

449 | A chord of length ( 16 mathrm{cm} ) is drawn in a circle of radius ( 10 mathrm{cm} . ) The distance of the chord from the centre of the circle is ( A cdot 8 mathrm{cm} ) B. ( 12 mathrm{cm} ) ( c cdot 6 c m ) D. ( 10 mathrm{cm} ) | 9 |

450 | Three wires of length ( l_{1}, l_{2}, l_{3} ) form a triangle surmounted by another circular wire, If ( l_{3} ) is the diameter and ( l_{3}=2 l_{1}, ) then the angle between ( l_{1} ) and ( l_{3} ) will be A ( .30^{circ} ) B. ( 60^{circ} ) ( c cdot 45^{circ} ) D. ( 90^{circ} ) | 9 |

451 | The distance, once around the circle is called A. diameter B. center c. circumference D. chord | 9 |

452 | Two chords of lengths ( 30 mathrm{cm} ) and ( 16 mathrm{cm} ) are on the opposite side of the centre of the circle. If the radius of the circle is 17 ( mathrm{cm}, ) find the distance between the chords. | 9 |

453 | ( A B ) and ( C D ) are two parallel chords of a circle such that ( A B=10 mathrm{cm} ) and ( C D=24 mathrm{cm} . ) If the chords are on the opposite sides of the centre and the distance between them is ( 17 mathrm{cm} ), the radius of the circle is A . ( 14 mathrm{cm} ) B. ( 10 mathrm{cm} ) c. ( 13 mathrm{cm} ) D. ( 15 mathrm{cm} ) | 9 |

454 | The line ( y=x ) is a tangent at (0,0) to a circle of radius is ( 1, ) then centre of the circle is ( ^{mathbf{A}} cdotleft(frac{1}{sqrt{2}}, frac{1}{sqrt{2}}right) ) B ( cdotleft(frac{1}{2 sqrt{2}}-frac{1}{sqrt{2}}right) ) ( ^{mathbf{c}} cdotleft(frac{-1}{sqrt{2}}, frac{1}{sqrt{2}}right) ) D. ( left(frac{-1}{sqrt{2}}, frac{-1}{sqrt{2}}right) ) | 10 |

455 | ( A B ) is chord of a circle with centre ( O ) and radius ( 17 mathrm{cm} . ) If ( O M perp A B ) and ( boldsymbol{O} boldsymbol{M}=mathbf{8} mathrm{cm} . ) The length of chord ( boldsymbol{A B} ) is A . ( 12 mathrm{cm} ) B. ( 30 mathrm{cm} ) c. ( 15 mathrm{cm} ) D. ( 24 mathrm{cm} ) | 9 |

456 | In the figure, if ( boldsymbol{A B}=boldsymbol{C D} ) and ( angle A O B=90^{circ} ) find ( angle C O D ) | 9 |

457 | In the given figure, ( O ) is the centre of a circle. If ( A B ) and ( A C ) are chords of the circle such that ( A B=A C ) and ( O P perp ) ( A B, O Q perp A C, ) prove that ( P B=Q C ) | 9 |

458 | Find the length of chord of circle with radius ( 5 c m ) and distance from center ( 2 c m ) | 9 |

459 | Find the value of ( x ) ( mathbf{A} cdot x=10 ) B. ( x=8 ) ( mathbf{c} cdot x=6 ) D. ( x=3 ) | 9 |

460 | i) A circle can have ( _{–} ) -parallel tangents. ii) The point common to the tangent and the circle is called | 10 |

461 | The circle and the square have the same center and the same area. If the circle has radius ( 1, ) the length of ( A B ) is A ( .4-7 ) B . ( 4-2 sqrt{pi} ) ( c cdot 2-sqrt{pi} ) D. ( sqrt{4-pi} ) | 9 |

462 | n the given figure, 0 is the centre of the circle. If ( angle B A D=75^{circ} ) and ( B C=C D ) find ( angle B O D, angle B C D, angle O B D ) | 9 |

463 | Two tangents PT and PT’ are drawn to a circle, with centre ( 0, ) from an external point P. Prove that ( angle mathrm{TPT}^{prime}=2 angle mathrm{OTT} ) ‘. | 10 |

464 | 23. A circle is drawn in a sector of a larger circle of radius r, as shown in figure. The smaller circle is tangent to the two bounding radii and the arc of the sector. The radius of the smaller circle is b. nie 60° | 10 |

465 | Find the value of ( x ) and ( y ) A ( cdot x=10^{circ}, y=7 ) B . ( x=18^{circ}, y=5 ) ( mathbf{C} cdot x=9^{o}, y=6 ) D. ( x=7^{circ}, y=6 ) | 9 |

466 | The centre of a circle touching two intersecting lines lies on the angle bisector of the lines. A. True B. False | 9 |

467 | ( A D ) is a diameter of a circle and ( A B ) is a chord. If ( boldsymbol{A} boldsymbol{D}=mathbf{3 4} boldsymbol{c m}, boldsymbol{A B}=mathbf{3 0} boldsymbol{c m} ) the distance of ( A B ) from the centre of the circle is ( A cdot 17 mathrm{cm} ) ( 3.15 mathrm{cm} ) ( c .4 mathrm{cm} ) ( 8.8 m ) | 9 |

468 | If ( A B ) is tangent to the circle at ( A ) and ( O B=13 mathrm{cm}, ) find the radius ( O A ) ( 4.5 mathrm{cm} ) ( 3.7 mathrm{cm} ) ( c cdot 8 c m ) ) | 10 |

469 | If the diameter of circle is ( 10 mathrm{cm}, ) then find the radius of circle. | 9 |

470 | The condition that the chord ( x cos alpha+ ) ( boldsymbol{y} sin boldsymbol{alpha}-boldsymbol{p}=mathbf{0} ) of ( boldsymbol{x}^{2}+boldsymbol{y}^{2}-boldsymbol{a}^{2}=mathbf{0} ) may subtend a right angle at the centre of the circle is A ( cdot a^{2}=2 p^{2} ) B ( cdot p^{2}=2 a^{2} ) c. ( a=2 p ) D. ( p=2 a ) | 9 |

471 | In the figure, line ( A B ) is a tangent to both the circles touching at ( A ) and ( B ) ( boldsymbol{O} boldsymbol{A}=mathbf{2 9}, boldsymbol{B P}=mathbf{1 8}, ) and ( boldsymbol{O P}=boldsymbol{6 1 .} ) The length of ( boldsymbol{A B} ) is ( A cdot 61 c m ) B. ( 60 mathrm{cm} ) c. ( 47 mathrm{cm} ) D. ( 11 c m ) | 10 |

472 | Find the radius of that circle whose area is ( 616 mathrm{cm}^{2}(text { in } mathrm{cm} .) ) | 9 |

473 | 67. Two circles touch internally at a point P and form a point T on the common tangent at P, tangent segments TQ, TR are drawn to the two circles then: (1) T9 = TR (3) TP_TR (2) TPP = 4TR (4) TP <TR | 10 |

474 | 70. A, B, C, D are four points on a circle. AC and BD intersect at a point E such that ZBEC = 130° and ZECD = 20°. ZBAC is (1) 120° (2) 90° (3) 100° (4) 110° | 9 |

475 | A circle of radius 7 is tangent to the lines of an angle ( 60^{circ} . ) is larger circle of radius ( r ) is tangent to same lines as well as given circle, then value of ( r ) is: begin{tabular}{l} A ( .7 sqrt{3} ) \ hline end{tabular} B. ( frac{28}{sqrt{3}} ) ( c cdot 21 ) D. 14 | 10 |

476 | ( boldsymbol{O} ) is the centre of the circle having radius ( 5 mathrm{cm} . A B ) and ( A C ) are two chords such that ( A B=A C=6 mathrm{cm} . ) If ( 0 mathrm{A} ) meets ( B C ) at ( M, ) then ( O M ) is equal to A . ( 3.6 mathrm{cm} ) B. ( 1.4 mathrm{cm} ) ( c cdot 2 c m ) ( 0.3 mathrm{cm} ) | 9 |

477 | 71. In the figure XAY is a tangent to the circle with centre O at A. If ZBAX=70°, ZBAQ = 40° then ZABO is equal to : UL (1) 20° (3) 35° (2) 30° (4) 40° | 9 |

478 | Find the centre and radius of the circle ( x^{2}+y^{2}-4 x-8 y-45=0 ) ( mathbf{A} cdot(2,6), sqrt{63} ) B . ( (2,4), sqrt{65} ) c. ( (2,-4), sqrt{66} ) D. None | 9 |

479 | The radius of the circle is ( 25 mathrm{cm} ) and the length of one of its chord is ( 40 mathrm{cm} ). find the distance of the chord from the centre | 9 |

480 | If the line ( 3 x-4 y-8=0 ) divides the circumference of the circle with centre (2,-3) in the ratio ( 1: 2 . ) Then, the radius of the circle is A. B. 2 ( c cdot 3 ) D. 4 | 9 |

481 | The tangent to the circle ( x^{2}+y^{2}=9 ) which is parallel to y-axis and does not lie in third quadrant, touches the circle at the point A. (-3,0) B. (3,0) D. (0,-3) | 10 |

482 | f’o’ is the centre of the circle ; ( 0 L=4 ) ( mathrm{cm}, mathrm{AB}=6 mathrm{cm} ) and ( mathrm{OM}=3 mathrm{cm}, ) then ( mathrm{CD} ) ( A cdot 4 mathrm{cm} ) B. ( 8 mathrm{cm} ) ( c cdot 6 mathrm{cm} ) D. ( 10 mathrm{cm} ) | 9 |

483 | ( A B ) and ( C D ) are two equal chords of a circle with centre ( boldsymbol{O} ) which intersect each other at right angle at point ( P . ) If ( O M perp A B ) and ( O N perp C D ; ) show that OMPN is a square | 9 |

484 | The length of chord of radius ( 25 mathrm{cm} ) and Distance at ( 7 mathrm{cm} ) is | 9 |

485 | In the diagram 0 is the centre of a circle. ( A E+E B=C E+E D O P perp A B ) and ( 0 Q ) ( perp ) CD then true relation between OP and OQ is A. ор > ( 0 Q ) B. op < ( 0 Q ) ( c cdot o p=frac{1}{2} o Q ) D. OP = OO | 9 |

486 | The figure is a circle with center ( O ) and diameter ( 10 mathrm{cm}, P Q=1 mathrm{cm} . ) Find the length of ( boldsymbol{R} boldsymbol{S} ) ( mathbf{A} cdot 6 mathrm{cm} ) B. ( 4 mathrm{cm} ) ( mathbf{c} .5 mathrm{cm} ) D. ( 3 mathrm{cm} ) | 9 |

487 | If the tangent ( P Q ) and ( P R ) are drawn to the circle ( x^{2}+y^{2}=a^{2} ) from the point ( Pleft(x_{1}, y_{1}right), ) then the equation of the circumcircle of ( triangle boldsymbol{P Q R} ) is A ( cdot x^{2}+y^{2}-x x_{1}-y y_{1}=0 ) B . ( x^{2}+y^{2}+x x_{1}+y y_{1}=0 ) c. ( x^{2}+y^{2}-2 x x_{1}-2 y y_{1}=0 ) D. None of these | 9 |

488 | Tangents drawn from the point ( boldsymbol{P}(mathbf{1}, boldsymbol{8}) ) to the circle ( x^{2}+y^{2}-6 x-4 y-11= ) 0 touch the circle at the point ( A ) and ( B ) The equation of the circumcentre of the ( triangle boldsymbol{P} boldsymbol{A} boldsymbol{B} ) is A ( cdot x^{2}+y^{2}+4 x-6 y+19=0 ) B . ( x^{2}+y^{2}-4 x-10 y+19=0 ) c. ( x^{2}+y^{2}-2 x+6 y-29=0 ) D. ( x^{2}+y^{2}+6 x-4 y+19=0 ) | 10 |

489 | The number of common tangents to the ( operatorname{circles} x^{2}+y^{2}=4 ) and ( x^{2}+y^{2}-4 x+ ) ( 2 y-4=0 ) is A . 1 B . 2 ( c .3 ) D. 4 | 10 |

490 | The circle whose radius is ( 1 mathrm{cm} ) then the diameter of the circle is | 9 |

491 | 55. In below figure O is centre of circle and ZAOB = 110° and ZAOC = 90°. then ZBAC will be equal to 1890 (1) 60 (3) 80 (2) 70 (4) 90° | 9 |

492 | Fill in the blanks with correct word(s) to make the statement true: A radius of a circle is a line segment with one end point at and the other end point on | 9 |

493 | f an isosceles ( triangle A B C ) in which ( A B= ) ( A C=6 mathrm{cm} ) is inscribed in a circle of radius ( 9 mathrm{cm} . ) Find area of the triangle. ( mathbf{A} cdot 8 c m^{2} ) B. ( 8 sqrt{2} c m^{2} ) ( c cdot 6 c m^{2} ) D. none | 9 |

494 | MN and MQ are tangents from a point ( mathrm{M} ) outside the given circle with center ( boldsymbol{O} ) If ( angle N O Q=120^{circ} ) then which of the following rlations holds true: A ( . N Q=M N=M Q ) в. ( N Q=O M ) c. ( O Q=O M ) D. ( O N=M N ) | 10 |

495 | It two equal chords of a circle intersect within the circle. Prove that the line joining the point of intersection to the centre makes equal angles with the chords. | 9 |

496 | If ( A B ) is a chord of a circle with centre ( O ) and ( P ) is a point on ( A B ) such that ( B P=4 P A, O P=5 mathrm{cm} ) and the radius of the circle is ( 7 mathrm{cm} ), find the value of ( (sqrt{6} times A B) ) | 9 |

497 | A chord of a circle of radius ( 12 mathrm{cm} ) subtends an angle of ( 120^{circ} ) at the center Find the area of the corresponding segment of the circle. | 9 |

498 | 67. The distance between two paral- lel chords of length 8 cm each in a circle of diameter 10 cm is (1) 6 cm (2) 7 cm (3) 8 cm (4) 5.5 cm | 9 |

499 | an infinite number of tangents can be drawn from (1,2) to the circle ( x^{2}+y^{2}- ) ( 2 x-4 y+lambda=0 ) then ( l a m b d a ) is A . -20 B. 0 c. 5 D. can no be determined | 10 |

500 | A circle of radius ( 3 mathrm{cm} ) can be drawn through two points ( A, B ) such that ( A B=6 mathrm{cm} ) State True or False ( A ). False B. True c. Cannot be determined D. None of the above | 9 |

501 | ( boldsymbol{X Y}=? ) 4.26 3.12 ( c cdot 14 ) ( D ) | 9 |

502 | If a line intersects a circle in two distinct points then it is known as a A. chord B. secant c. tangent D. segment | 10 |

503 | 57. AB and CD (AB||CD) are the two chord of a circle with length 5 cm and 11 cm respectively. If the distance between AB and CD is 3 cm, then the radius of circle will be (1) 1104 cm (2) 194 cm cm cm | 9 |

504 | f the line ( x cos alpha+y sin alpha=p ) represents the common chord ( A P Q B ) of the circles ( x^{2}+y^{2}=a^{2} ) and ( x^{2}+y^{2}= ) ( b^{2}(a>b) ) as shown in the figure, then ( A P ) is equal to A ( cdot sqrt{a^{2}+p^{2}}+sqrt{b^{2}+p^{2}} ) B. ( sqrt{a^{2}-p^{2}}+sqrt{b^{2}-p^{2}}^{2} ) c. ( sqrt{a^{2}-p^{2}}-sqrt{b^{2}-p^{2}} ) D. ( sqrt{a^{2}+p^{2}}-sqrt{b^{2}+p^{2}}^{2} ) | 9 |

505 | A regular hexagon & a regular dodecagon are inscribed in the same circle. If the side of the dodecagon is ( (sqrt{3}-1), ) then the side of the hexagon is A ( cdot sqrt{2}+1 ) B. ( frac{sqrt{3}+1}{2} ) c. 2 D. ( sqrt{2} ) | 9 |

506 | Which of the following is secant to the circle given above? ( A cdot A B ) B. CD ( c cdot c ) ( D . P O ) | 10 |

507 | The line segment joining any two points on a circle is called a or an A. arc of the chord B. radius of the circle c. chord of the circle D. tangent of the circle | 9 |

508 | The radius of a circle is ( 17.0 mathrm{cm} ) and the length of perpendicular drawn from its centre to a chord is ( 8.0 mathrm{cm} . ) Calculate the length of the chord. | 9 |

509 | Explain the followings: Chord | 9 |

510 | Chord ( A B ) of the circle ( x^{2}+y^{2}=100 ) passes through the point (7,1) and subtends an angle of ( 60^{circ} ) at the circumference of the circle. If ( m_{1} ) and ( m_{2} ) are the slopes of two such chords then the value of ( m_{1} m_{2}, ) is A . -1 B. 1 c. ( frac{7}{12} ) D. – 3 | 9 |

511 | Line ( 3 x-4 y=k ) will cut the circle ( x^{2}+y^{2}-2 x+4 y-11=0 ) at distinct points, if A ( cdot k>frac{25}{7} ) в. ( 15<k<30 ) c. ( -9<k<31 ) D. None of these | 10 |

512 | In a circle whose radius is 10 inches, a chord is 6 inches from the center. What is the length of the chord? A. 4 inches B. 6 inches c. 8 inches D. 16 inches | 9 |

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.

**help, strategies and expertise**.