## Circles

**Chapter 10**

**Exercise 10.3**

**EX 10.3 QUESTION 1.**

**Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?**

**Solution:**

(i) Zero

(ii) One

(iii) One

(iv) Two

The maximum number of common points is two.

**EX 10.3 QUESTION 2.**

** Suppose you are given a circle. Give the construction to find its centre.**

**Solution:**

Construction:

- Join PR and OR
- Draw a perpendicular bisector of PR and QR which intersect at point O.
- Taking O as a centre and OP as a radius.

**EX 10.3 QUESTION 3.**

**If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.**

**Solution:**

To Prove: Points P and Q lie on the perpendicular bisector of common chord AB.

Construction: Join point P and Q to midpoint M of chord AB.

Proof: AB is a chord of a circle C, PM is a bisector of chord AB.

∴ PM ⊥ AB [Line drawn through the centre of the circle to bisect a chord is perpendicular to the chord]

∠PMA = 90°

Now, AB is a chord of circle C, and QM is a bisector of chord AB.

∴ QM ⊥ AB [Line drawn through the centre of the circle to bisect a chord is perpendicular to the chord]

∠QMA = 90°

∠PMA + ∠QMA= 90° + 90° = 180°

∠PMA and ∠QMA forming linear pair. So PMQ is a straight line and Points P and Q lie on the perpendicular bisector of common chord AB.