# Circles (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.