Hyperbola

Given, equation of circle $\Rightarrow x^{2}+y^{2}=25$ Equation of hyperbola $\Rightarrow \; \frac{x^{2}}{9} - \; \frac{y^{2}}{16} = 1$ Let the mid-point of the chord of the circle be $(h, k)$. The equation of a chord through its mid-point is simply $T=S_{1}$ where $T$ is the equation of tangent and $S_{1}$ is the value of $S$ by putting $(h, k)$. Here, $\; \; h x + k y = h^{2} + k^{2}$ The equation of tangent to hyperbola $\; \frac{x^{2}}{a^{2}} - \; \frac{y^{2}}{b^{2}} = 1$ is $y = m x \pm \sqrt{a^{2} m^{2} - b^{2}}$ Here, $y = m x \pm \sqrt{9 m^{2} - 16}$ So, $y = \; \frac{-h x}{k} + \; \frac{h^{2}+k^{2}}{k}$ .....(i) and $y = m x \pm \sqrt{9 m^{2} - 16}$.....(ii) Eqs. (i) and (ii) are identical, so $\; \frac{1}{1} = \; \frac{-\frac{h}{k}}{m} = \; \frac{h^{2}+k^{2}}{k \sqrt{9 m^{2}-16}} \Rightarrow m = \; \frac{-h}{k}$ $\Rightarrow 9 m^{2} - 16 = \; \frac{(h^{2}+k^{2})^{2}}{k^{2}} \Rightarrow 9 \left( \; \frac{h^{2}}{k^{2}} \right) - 16 = \; \frac{(h^{2}+k^{2})^{2}}{k^{2}}$ $\Rightarrow 9 h^{2} - 16 k^{2} = (h^{2}+k^{2})^{2}$ $\left\{ \begin{array}{l} h \rightarrow x \\ k \rightarrow y \end{array} \right.$ So,