Chapter 10: Circles
CBSE Official PYQs (2015-2026)
Q1. If tangents \(PA\) and \(PB\) from a point \(P\) to a circle with centre \(O\) are inclined to each other at an angle of \(80^\circ\), then \(\angle POA\) is equal to :
Q2. Let \(PQ\) be a chord of a circle and \(PT\) be the tangent at \(P\) such that \(\angle QPT = 60^\circ\). Then \(\angle PRQ\) (where R is a point on the major arc) is equal to :
Q3. If the angle between two radii of a circle is \(130^\circ\), then the angle between the tangents at the ends of the radii is :
Q4. Let \(PA\) and \(PB\) be tangents to a circle with centre \(O\). If \(\angle APB = 60^\circ\), then \(\angle OAB\) is :
Q5. A quadrilateral \(ABCD\) is circumscribed to a circle with centre \(O\). If \(AD = 23\text{ cm}\), \(AB = 29\text{ cm}\), \(DS = 5\text{ cm}\) (where \(S\) is the point of contact on \(AD\)) and \(\angle B = 90^\circ\), then the radius of the circle is :
Q6. From a point \(P\) which is at a distance of \(13\text{ cm}\) from the centre \(O\) of a circle of radius \(5\text{ cm}\), the pair of tangents \(PQ\) and \(PR\) to the circle are drawn. Then the area of the quadrilateral \(PQOR\) is :
Q7. If two tangents inclined at an angle of \(60^\circ\) are drawn to a circle of radius \(3\text{ cm}\), then length of each tangent is equal to :
Q8. Let \(AB\) and \(AC\) be tangents to a circle with centre \(O\) from an external point \(A\). If \(\angle BOC = 140^\circ\), then \(\angle BAC\) is equal to :
Q9. Two concentric circles are of radii \(10\text{ cm}\) and \(6\text{ cm}\). The length of the chord of the larger circle which touches the smaller circle is :
Q10. Let \(O\) be the centre of a circle, \(AB\) be a chord and \(AT\) be the tangent at \(A\). If \(\angle AOB = 100^\circ\), then \(\angle BAT\) is equal to :
Q11. How many tangents can a circle have at any point on its boundary?
Q12. Let \(PA\) and \(PB\) be tangents to a circle with centre \(O\) from an external point \(P\). If \(\angle AOB = 115^\circ\), then \(\angle APB\) is :
Q13. If a circle touches all the four sides of a quadrilateral \(ABCD\) at points \(P, Q, R\) and \(S\) respectively, then :
Q14. Let \(O\) be the centre of a circle. A line \(PQ\) is tangent to the circle at \(A\). If \(B\) is a point on the circle such that \(\angle OAB = 30^\circ\), then the measure of \(\angle ABQ\) is :
Q15. A line intersecting a circle in two distinct points is called a :
Q16. Let \(TA\) and \(TB\) be two tangents to a circle centred at \(O\) from an external point \(T\). If \(\angle OAB = 30^\circ\), then the measure of \(\angle ATB\) is :
Q17. The maximum number of parallel tangents a circle can have is :
Q18. If the perimeter of a protractor is \(72\text{ cm}\), then its radius is (taking \(\pi = 22/7\)) :
Q19. Two concentric circles have a common centre \(O\). A chord \(AB\) of the larger circle touches the smaller circle at \(C\). If \(AB = 8\text{ cm}\), then the length of \(AC\) is :
Q20. If tangent \(PQ\) at a point \(P\) of a circle of radius \(5\text{ cm}\) meets a line through the centre \(O\) at a point \(Q\) so that \(OQ = 12\text{ cm}\), then the length of \(PQ\) is :
Q21. A quadrilateral \(ABCD\) circumscribes a circle. If \(AB = 6\text{ cm}\), \(BC = 7\text{ cm}\) and \(CD = 4\text{ cm}\), then the length of \(AD\) is :
Q22. Assertion (A): If the length of tangent from an external point to a circle is 8 cm, and radius of the circle is 6 cm, then distance of point from centre is 10 cm.
Reason (R): Tangent at any point of a circle is perpendicular to the radius through the point of contact.
Q23. Let \(O\) be the centre of a circle and \(PT\) be a tangent to the circle at \(T\). If \(OP = 13\text{ cm}\) and \(PT = 12\text{ cm}\), then the radius of the circle \(OT\) is :
Q24. If two tangents are inclined at an angle of \(120^\circ\) to each other are drawn to a circle of radius \(4\text{ cm}\), then the length of each tangent is :
Q25. Tangents \(PA\) and \(PB\) are drawn to a circle centered at \(O\) from an external point \(P\). If \(\angle APB = 50^\circ\), then what is the measure of \(\angle AOB\)?
Q26. Let \(PQ\) and \(PR\) be two tangents drawn to a circle with centre \(O\) and radius \(5\text{ cm}\). If \(\angle QOR = 120^\circ\), then the length of \(PQ\) is :
Q27. A circle circumscribes a rectangle \(ABCD\). If \(AB = 8\text{ cm}\) and \(BC = 6\text{ cm}\), then the radius of the circle is :
Q28. Let \(PA\) be a tangent to a circle centred at \(O\) from an external point \(P\). If \(OA = 8\text{ cm}\) and \(OP = 17\text{ cm}\), then the length of the tangent \(PA\) is :
Q29. If the radii of two concentric circles are \(13\text{ cm}\) and \(12\text{ cm}\), then the length of the chord of the outer circle which touches the inner circle is :
Q30. A quadrilateral \(PQRS\) circumscribes a circle. If \(PQ + RS = 18\text{ cm}\), then the perimeter of the quadrilateral \(PQRS\) is :
Q31. Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
Q32. A circle touches all the four sides of a quadrilateral \(ABCD\). If \(AB = 8\text{ cm}\), \(BC = 9\text{ cm}\) and \(CD = 6\text{ cm}\), find the length of \(AD\).
Q33. If from an external point \(P\), tangents \(PA\) and \(PB\) are drawn to a circle centred at \(O\), and \(\angle APB = 120^\circ\), prove that \(OP = 2AP\).
Q34. Let \(PA\) and \(PB\) be tangents to a circle from an external point \(P\). If \(OP\) is equal to the diameter of the circle, show that \(\Delta APB\) is an equilateral triangle.
Q35. Find the HCF and LCM of 306 and 657, and then find the radius of a circle whose circumference is numerically equal to the HCF of these two numbers (taking \(\pi = 22/7\)).
Q36. Two tangents \(PQ\) and \(PR\) are drawn to a circle with centre \(O\) from an external point \(P\). If \(\angle QPR = 80^\circ\), then find the measure of \(\angle QOR\) and \(\angle OQR\).
Q37. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Q38. Let \(XP\) and \(XQ\) be two tangents to a circle with centre \(O\) from an external point \(X\). Let \(ARB\) be another tangent touching the circle at \(R\) (where \(A\) lies on \(XP\) and \(B\) lies on \(XQ\)). If \(XP = 16\text{ cm}\), find the perimeter of \(\Delta XAB\).
Q39. If \(PQ\) is a tangent to a circle of radius \(8\text{ cm}\) from an external point \(P\) such that \(OP = 17\text{ cm}\), find the length of \(PQ\).
Q40. Two circles touch each other externally at \(C\). Let \(PT\) be a common tangent touching the circles at \(P\) and \(T\) respectively, and let the tangent at \(C\) intersect \(PT\) at \(Q\). Show that \(\angle PCT = 90^\circ\).
Q41. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
Q42. Let \(AB\) be a chord of length \(8\text{ cm}\) of a circle of radius \(5\text{ cm}\). The tangents at \(A\) and \(B\) intersect at a point \(P\). Find the length of the tangent \(PA\).
Q43. Prove that the length of tangents drawn from an external point to a circle are equal.
Q44. Let \(xy\) and \(x'y'\) be two parallel tangents to a circle with centre \(O\) and let another tangent \(AB\) with point of contact \(C\) intersect \(xy\) at \(A\) and \(x'y'\) at \(B\). Prove that \(\angle AOB = 90^\circ\).
Q45. Prove that a parallelogram circumscribing a circle is a rhombus.
Q46. Let \(PQ\) be a chord of length \(16\text{ cm}\) of a circle of radius \(10\text{ cm}\). The tangents at \(P\) and \(Q\) intersect at a point \(T\). Find the length of the tangent \(TP\).
Q47. A triangle \(ABC\) is drawn to circumscribe a circle of radius \(4\text{ cm}\) such that the segments \(BD\) and \(DC\) into which \(BC\) is divided by the point of contact \(D\) are of lengths \(8\text{ cm}\) and \(6\text{ cm}\) respectively. Find the sides \(AB\) and \(AC\).
Q48. Let \(PQ\) be a tangent to a circle centered at \(O\). If \(OP = 15\text{ cm}\) and \(PQ = 12\text{ cm}\), and \(QR\) is a diameter of the circle, find the length of \(PR\).
Q49. Prove that the lengths of tangents drawn from an external point to a circle are equal. Using the above result, prove that in a triangle \(ABC\) circumscribing a circle of radius \(r\), the area is given by \(\text{Area} = r \times s\), where \(s\) is the semi-perimeter of the triangle.
Q50. Tangents \(PQ\) and \(PR\) are drawn from an external point \(P\) to a circle with centre \(O\). If chord \(QR\) is parallel to tangent \(XY\) drawn at point \(S\) of the circle, prove that point \(S\) bisects the arc \(QSR\).
Q51. Case Study: Roller Coaster Track Design
A designer is planning a circular loop for a new roller coaster track. For structural stability, support beams are attached as tangents from an external control deck \(P\) to the circular loop at points \(A\) and \(B\). The control deck \(P\) is located \(25\text{ m}\) away from the centre \(O\) of the circular loop. The radius of the loop is \(15\text{ m}\).
Based on the above information, answer the following questions:
- Find the length of each support beam (tangent) \(PA\) and \(PB\). (2 Marks)
- Find the total area of the quadrilateral \(PAOB\) formed by the tangents and the radii. (2 Marks)
(i) \(PA = PB = 20\text{ m}\).
(ii) Area of \(PAOB = 300\text{ sq. m}\).
Q52. Case Study: Satellite Telecommunications Orbit
A telecommunications satellite \(S\) is orbiting the Earth at a constant altitude. The signals from the satellite \(S\) touch the surface of the Earth (modelled as a circle with centre \(O\) and radius \(6400\text{ km}\)) at points \(A\) and \(B\), which represent the horizon limits of the satellite's coverage. The angle \(ASB\) between the two tangent signal lines is \(60^\circ\).
Based on the above information, answer the following questions:
- Find the distance of the satellite \(S\) from the centre of the Earth \(O\). (2 Marks)
- Find the height of the satellite above the surface of the Earth. (2 Marks)
(i) Distance \(OS = 12800\text{ km}\).
(ii) Height = \(OS - OA = 12800 - 6400 = 6400\text{ km}\).