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.