Question

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

Answer 1

Let $MN$ be common chord of the two circles, such that when $MN$ is extended to $O$, $OP$ and $OQ$ are two tangents on the respective circles.

Now, for the first circle, as $ON$ is the secant,

$\Rightarrow O{P}^2=ON\times OM$..(i)

Similarly, for the second circle, as $ON$ is the secant,

$\Rightarrow O{Q}^2=ON\times OM$..(ii)

From (i) and (ii),

$\Rightarrow O{P}^2=O{Q}^2$

$\therefore OP=OQ$