1. How many tangents can a circle have?
2. Fill in the blanks:
(i) A tangent to a circle intersects it in_______ point(s).
(ii) A line intersecting a circle in two points is called a _____.
(iii) A circle can have________ parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called_____.
3. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is:
(A) 12 cm (B) 13 cm (C) 8.5 cm (D) √119 cm
4. Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
In Q.5 to 7, choose the correct option and give justification.
5. 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:
(A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5 cm
6. In figure, if TP and TQ are the two tangents to a circle with centre 0 so that ∠ POQ = 1100, then PTQ is equal to:
(A) 600 (B) 700 (C) 800 (D) 900
7. If tangents PA and PB from a point P to a circle with centre 0 are inclined to each other at angle of 800, then ∠ POA is equal to:
(A) 500 (B) 600 (C) 700 (D) 800
8. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
9. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
10. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
11. 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.
12. A quadrilateral ABCD is drawn to circumscribe a circle (see figure). Prove that: AB + CD = AD + BC
13. In figure, XY and X’Y’ are two parallel tangents to a circle with centre 0 and another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that ∠ AOB = 900.
14. Prove that the angel 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.
15. Prove that the angel 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
16. Prove that the parallelogram circumscribing a circle is a rhombus.
17. 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 (see figure). Find the sides AB and AC.
18. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.