Circles

Exercise 1

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.

Exercise 2

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.