1. prove that two different circles cannot intersect each other at more thantwo points.

Text Solution

Solution : To prove: Two distinct circles cannot intersect each other in more than two points.<br> Proof: Suppose that two distinct circles intersect each other in more than two points.<br>So, These points are non-collinear points.<br> Three non-collinear points determine one and only one circle.<br> Since there should be only one circle. Therefore, from those three points, 2 circles cannot pass.<br> This contradicts the given, which shows that our assumption is wrong.<br> Hence, two distinct circles cannot intersect each other in more than two points.

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Prove that two different circles cannot intersect each other at more than two points.

Solution

To prove: Two different circles cannot intersect each other at more than two points.

Construction: Let two circles intersect each other at three points $A,B$ and $C$

Proof :

Since two circles with centres $O$ and $O^{\prime }$ intersect at $A,B$ and $C$.

$\therefore A,BandC$ are non-collinear points

$\therefore$ Circle with centre $O$ passes through three points $A,B$ and $C$.

and circle with centre $O^{\prime }$ also passes through three points $A,B$ and $C$.

But one and only one circle can be drawn through three points

$\therefore$ Our supposition is wrong

$\therefore$ Two circle cannot intersect each other not more than two points.

Observe the following figure,

Mathematics

RD Sharma

Standard IX