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Question

$$A, B, C$$ are the centres of three circles which cut each other orthogonally. The radical centre of the three circles is .......... of the $$\Delta$$ $$ABC$$


A
Incentre
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B
Centroid
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C
Orthocentre
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D
Circumcentre
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Solution

The correct option is C Orthocentre

Consider three circles AB and C, no two of which are concentric. The radical axis theorem states that the three radical axes (for each pair of circles) intersect in one point called the radical center, or are parallel.  In technical language, the three radical axes are concurrent  (share a common point); if they are parallel, they concur at a point of infinity.

A simple proof is as follows. The radical axis of circles A and B is defined as the line along which the tangents to those circles are equal in length $$a =b$$. Similarly, the tangents to circles B and C must be equal in length on their radical axis. By the transitivity of equality, all three tangents are equal $$a=b=c$$ at the intersection point r of those two radical axes. Hence, the radical axis for circles A and C must pass through the same point r, since $$a=c$$ there. This common intersection point r is the radical center.

There is a unique circle with its center at the radical center that is orthogonal to all three circles. This follows, also by transitivity, because each radical axis, being the locus of centers of circles that cut each pair of given circles orthogonally, requires all three circles to have equal radius at the intersection of all three axes


Maths

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