Find the area of the triangle formed by the tangents from the point $(h, k)$ to the circle $x^{2} + y^{2} = a^{2}$ and their chord of contact.

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Solution

Equation of the chord of contact $A B$ of tangents drawn from the point $P (h, k)$ to the given circle is $h x + k y - a^{2} = 0$ .
$O M =$ perp. distance of $(0, 0)$ from $A B$ .
$∴ O M = \frac{- a^{2}}{\sqrt{(h^{2} + k^{2})}}$
$∴ A M^{2} = O A^{2} - O M^{2} = a^{2} - \frac{a^{4}}{h^{2} + k^{2}}$
$= \frac{a^{2} (h^{2} + k^{2} - a^{2})}{(h^{2} + k^{2})}$
$∴ A B = 2 A M = 2 a \sqrt{(\frac{h^{2} + k^{2} - a^{2}}{h^{2} + k^{2}})}$
$P M =$ perp. distance of $(h, k)$ from $A B$
$∴ P M = \frac{h^{2} + k^{2} - a^{2}}{\sqrt{(h^{2} + k^{2})}}$
$△ P A B = \frac{1}{2} P M . A B$
$= \frac{1}{2} . \frac{h^{2} + k^{2} - a^{2}}{\sqrt{(h^{2} + k^{2})}} . 2 a \sqrt{(\frac{h^{2} + k^{2} - a^{2}}{h^{2} + k^{2}})}$
$= a \frac{(h^{2} + k^{2} - a^{2})^{3 / 2}}{h^{2} + k^{2}} . . . . . (2)$ .

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