Explanation

(d)
Equation of tangent to $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is
$y = mx + \sqrt{a^{2}m^{2} + b^{2}}$
And equation of tangent to $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 2$ is
$y = mx + \sqrt{2a^{2}m^{2} + 2b^{2}}$
For common tangent,
$a^{2}m^{2} + b^{2} = 2a^{2}m^{2} - 2b^{2}$
$\Rightarrow a^{2}m^{2} = 3b^{2}\ \Rightarrow m = \pm \frac{\sqrt{3}b}{a}$
$\therefore$ Equation of common tangent is $y = \frac{\sqrt{3}b}{a}x + 2b.$