Question
If the tangent at the point $P(h,k)$ on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ cuts the circle $x^{2} + y^{2} = a^{2}$ at the points $Q\left( x_{1},y_{1} \right)$ and $R\left( x_{2},y_{2} \right)$ then $\frac{1}{y_{1}} + \frac{1}{y_{2}}$ is equal to:
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Solution
Sol. (a) Equation of tangent at $P$ is $\frac{xh}{a^{2}} - \frac{yk}{b^{2}} = 1$
Put $x = \left( 1 + \frac{yk}{b^{2}} \right)\frac{a^{2}}{h}\ $ in equation $x^{2} + y^{2} = a^{2}$
$$\Rightarrow \frac{1}{y_{1}} + \frac{1}{y_{2}} = \frac{\frac{- 2ka^{4}}{b^{2}h^{2}}}{\frac{a^{4} - a^{2}h^{2}}{h^{2}}}$$
$$\begin{matrix} & \ = - \frac{2ka^{2}}{b^{2}\left( a^{2} - h^{2} \right)} = - \frac{2k}{b^{2}\left( 1 - \frac{h^{2}}{a^{2}} \right)} \\ & \ = - \frac{2k}{b^{2}\left( - \frac{k^{2}}{b^{2}} \right)} = \frac{2}{k} \end{matrix}$$
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