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

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 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.


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