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If from any point on the common chord of two intersecting circles, tangents be drawn to the circles, prove that they are equal.
Given:
From any point on the common chord of two intersecting circles, tangents be drawn to the circles.
To do:
We have to prove that they are equal.
Solution:
Let $QR$ be the common chord of two circles intersecting each other at $Q$ and $R$.
$P$ be the point on $QR$ when produced.
$PY$ and $PX$ be the tangents drawn to the circles with centres $O$ and $C$ respectively.
Proof:
$PY$ is the tangent and $PQR$ is the secant to the circle with centre $O$.
This implies,
$PY^2 = PQ \times PR$...….(i)
Similarly,
$PX$ is the tangent and $PQR$ is the secant to the circle with centre $C$.
This implies,
$PX^2 = PQ \times PR$..….(ii)
From (i) and (ii), we get,
$PY^2 = PX^2$
$PY = PX$
Hence proved.
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