Question

Sol.(d) $\frac{x^{2}}{5} + \frac{y^{2}}{3} = 1$

Equation of director circle is

$$\begin{array}{r} x^{2} + y^{2} = 8\#(1) \end{array}$$

$(\sqrt{7},1)$ satisfy the equation (1)

⇒ Angle between tangents is $\frac{\pi}{2}$

Let any tangent to the ellipse is $y = mx \pm \sqrt{a^{2}m^{2} + b^{2}}\ $ which is passing through $(\sqrt{7},1)$

$$\begin{matrix} (1 - \sqrt{7}m)^{2} & \ = 5m^{2} + 3 \\ \Rightarrow \ m^{2} - \sqrt{7}m - 1 & \ = 0 \end{matrix}$$

If $m_{1},m_{2}$ are

roots $\Rightarrow m_{1}m_{2} = - 1$

⇒ two tangents are perpendicular to each other.