Q1. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is:
Q2. If TP and TQ are the two tangents to a circle with centre O so that $\angle POQ = 110^\circ$, then $\angle PTQ$ is equal to:
Q3. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. The radius of the circle is:
Q4. Assertion (A): If a line intersects a circle at two distinct points, it is called a secant. Reason (R): A tangent to a circle intersects the circle at exactly one point.
Q5. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Solve on white paper.
Q6. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Q7. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that $AB + CD = AD + BC$.
Q8. 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.
Q9. 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 $\angle AOB = 90^\circ$.
Q10. Prove that the lengths of tangents drawn from an external point to a circle are equal. Using this, show that if a parallelogram circumscribes a circle, it is a rhombus.
Q11. Case Study:
A circular park has a radius of 20 m. Two straight paths, which are tangents to the circular boundary of the park, meet at a point P outside the park. The distance of point P from the centre of the park is 50 m.
(i) Find the length of each path (tangent) from point P to the point of contact with the park.
(ii) If the two paths are inclined to each other at an angle of $60^\circ$ instead, find the distance of P from the centre.
(iii) What is the perimeter of the quadrilateral formed by the two tangents and the two radii joining the points of contact?