Question

Sol. (b) Equation of tangent to hyperbola is

$$y = mx \pm \sqrt{a^{2}m^{2} - b^{2}}$$

∵ It touches parabola also Equation of tangent to parabola.

$$\begin{matrix} \Rightarrow & a^{2}m^{2} - b^{2} = \frac{a^{2}}{m^{2}} \\ \Rightarrow & a^{2}m^{4} - b^{2}m^{2} - a^{2} = 0 \\ & m^{2} = \frac{b^{2} \pm \sqrt{b^{4} + 4a^{4}}}{2a^{2}} \\ & \ = \frac{1}{2}\left\lbrack \frac{b^{2}}{a^{2}} \pm \sqrt{\frac{b^{4}}{a^{4}} + 4} \right\rbrack \end{matrix}$$

Clearly $\frac{b^{2}}{a^{2}} \neq 1,m^{2} > 1$

$$\Rightarrow \ m \in ( - \infty, - 1) \cup (1,\infty) $$and $m^{2} \neq (1 + \sqrt{5})/2$