Question

Sol.(b) Equation of tangent to $\frac{x^{2}}{a^{2} + b^{2}} + \frac{y^{2}}{b^{2}} = 1$ is

$$\begin{array}{r} y = mx \pm \sqrt{\left( a^{2} + b^{2} \right)m^{2} + b^{2}}\#(1) \end{array}$$

∵ (1) is also tangent to $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{a^{2} + b^{2}} = 1$

$${\Rightarrow \left( a^{2} + b^{2} \right)m^{2} + b^{2} = a^{2}m^{2} + \left( a^{2} + b^{2} \right) }{\Rightarrow m = \pm \frac{a}{b} }$$Equation of common tangents are

$$by = ax \pm \sqrt{a^{2}\left( a^{2} + b^{2} \right) + b^{4}}$$