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Question

If from any point on the common chord of two intersecting circles, tangents be drawn to the circles, prove that they are equal.

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Solution

Let 2 circle intersect at points A and B such that AB is the common chord. Extend the common chord to point P.Let PR and PQ be two tangents respectively on the two circles (as shown in the figure).
Since PQ is that tangent and PAB is the secant for one circle,
PQ2=PB×PA...(i)
Similarly, PR is the tangent and PAB is the secant for the other circle,
PR2=PB×PA...(ii)
From (i) and (ii),
PQ2=PR2
PQ=PR
Hence, the tangents drawn from any point of the chord onto circles are equal.

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