Question

If two circle intersect at two point prove that their centres lies on the perpendicular bisector of the common chord

(class9)

Answer 1

Let there be two circles of centres O and P which intersect at two points A and B.

To prove:- OP is perpendicular to AB

Proof

let OC is a line which passes through OC and is perpendicular to AB

So, C is the mid point of AB............(1)  (theorem is given in NCERT book)

Similarly, PD is a line perpendicular to AB

So, D is the mid point of AB............(2)

Now

Comparing equation 1 and 2

C and D both are mid points of AB

this is only possible when C coincide on D.

Therefore, OP is the perpendicular bisector of AB. (OC+PD=OP and OC perpendicular to AB and PD perpendicular to AB)

Hence, proved.

Answer 2

Hello mate =_=

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Solution:

Construction:

1) Draw two circles with centres O and O'.

2)Join A and B to get a common chord AB.

3) Join O and O' with the mid-point M of AB.

To prove: Centres lie on the perpendicular bisector of the common chord. In other words, we need to prove that OO' is a straight line and ∠AMO=∠AMO′=90°

In △AOB, M is the mid-point of chord AB.

⇒∠AMO=90°        .....(1)

(The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.)

Similarly, in △AO′B, M is the mid-point of chord AB.

⇒∠AMO′=90°        .......(2)

(The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.)

hope, this will help you.

Thank you______❤

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