Circles (PYQs)
1. The lengths of tangents drawn from an external point to a circle are equal.
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2. If two tangents inclined at an angle 60° are drawn to a circle of radius 3cm, then find the length of each tangent.
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3. Two concentric circles are of radii 5cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
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4. Prove that the parallelogram circumscribing a circle is a rhombus.
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5. In the figure given, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that –AOB = 90°.
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6. In the given figure, OP is equal to diameter of the circle. Prove that ABP is an equilateral triangle.
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7. In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC.
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8. In the given figure, from an external point P, two tangents PQ and PR are drawn to a circle of radius 4 cm with centre O. If ∠QPR = 90°, then length of PQ is
(a) 3 cm (b) 4 cm (c) 2 cm (d) 2√2 cm
View AnswerAns. (b) 4 cm
9. Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle.
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10. From an external point P, tangents PA and PB are drawn to a circle with centre O. If ∠PAB = 50°, then find ∠AOB.
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11. If the angle between two tangents drawn from an external point ‘P’ to a circle of radius ‘r’ and centre O is 60°, then find the length of OP.
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12. If the radii of two concentric circles are 4 cm and 5 cm, then find the length of each chord of one circle which is tangent to the other circle.
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13. If PQ = 28 cm, then find the perimeter of DPLM.
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14. PQ is a tangent to a circle with centre O at point P. If DOPQ is an isosceles triangle, then find ∠OQP
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15. Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.
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16. In the given figure, from an external point P, two tangents PT and PS are drawn to a circle with centre O and radius r. If OP = 2r, show that ∠OTS = ∠OST = 30°.
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17. In the given figure, AP and BP are tangents to a circle with centre O, such that AP = 5cm and –APB = 60°. Find the length of chord AB
Ans. 5 cm
18. Prove that the tangents at the extremities of a chord of a circle make equal angles with the chord.
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19. In the given figure, common tangents AB and CD to two circles with centres O1 and O2 intersect at E. Prove that AB = CD.
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20. If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that –QPR = 120°, prove that 2PQ = PO
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21. In the given figure, a circle inscribed in DABC, touches its sides BC, CA and AB at the points P, Q and R respectively. If AB = AC, then prove that BP = CP.
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22. In the given figure, DABC is circumscribing a circle. Find the length of BC.
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23. A circle is inscribed in a ∆ABC having sides 8 cm, 10 cm and 12 cm as shown in the following figure. Find AD, BE and CF.
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24. PA and PB are tangents from P to the circle with centre O. At point M, a tangent is drawn cutting PA at K and PB at N. Prove that KN = AK + BN.
View Answer25. In the given figure, O is the centre of the circle. Determine √AQB and √AMB, if PA and PB are tangents and √APB = 75°
Ans. . √AQB = 52.5°, √AMB = 127.5°
26. In the given figure, AB is the diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58°, find ∠ATQ
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27. From a point T outside a circle of centre O, tangents TP and TQ are drawn to the circle. Prove that OT is the right bisector of the line segment PQ.
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28. In the given figure, two tangents RQ and RP are drawn from an external point R to the circle with centre O. If ∠PRQ = 120°, then prove that OR = PR + RQ.
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29. In the given figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If ∠TPQ = 70°, find ∠TRQ
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30. In the given figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q. If PA = 12 cm, QC = QD = 3 cm, then find PC + PD.
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31. A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA
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32. In the given figure, find the perimeter of DABC, if AP = 12 cm
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33. In the given figure, a circle touches all the four sides of a quadrilateral ABCD in which AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD.
Ans. 3 cm
34. In the given figure, two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2 ∠OPQ
35. In the figure, quadrilateral ABCD is circumscribing a circle with centre O and AD ^ AB. If radius of incircle is 10 cm, then find the value of x
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36. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.
