# Circles – Exercise 10.3 – (MATHEMATICS) – 9th Class

** Exercise 10.3**

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

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**Sol.**

(i)

NO point is common

(ii)

One point P is Common.

(iii)

One point P is common.

(iv)

Two points P and Q are common.

Hence, the maximum number of common points is two, which is incase (iv).

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

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**Sol.**Steps to find centre of the circle:

Two non-parallel chords AB and CD of a circle are drawn.

(ii) Perpendicular bisectors of AB and CD are drawn.

(iii) Let these bisectors meet at O.

Then 0 is the required centre of the circle.

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

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**Sol.**Let two circles with centres 0 and 0′ intersect each other at P and Q. Thus PQ is the common chord as shown in the adjoining figure.

Let us draw perpendicular OL on PQ,

then OL bisects PQ at L,

[Perpendicular from centre of a circle to the chord bisects the chord]

i.e., ∠OLP = 90° and PL = QL ……(i)

L and 0′ are joined

Then O’L is perpendicular to PQ …(ii)

[Line segment joining the centre of the circle to midpoint of the chord is perpendicular to the chord]

From (i) and (ii), we have

∠OLP + ∠O’LP = 90° + 900 = 180′

⇒ ∠OLP and ∠O’LP from a linear pair.

Hence, OLO’ is a straight line

with PL = QL.

Hence centres of the two circles lie on the perpendicular bisector of the common chord.