Explanation

(d)
The equation of tangent to the parabola
$y^{2} = 4x\ is\ y = mx + \frac{1}{m}$
This is also the tangent to parabola $x^{2} = - 8y$
$\therefore\ x^{2} = - 8\left( mx + \frac{1}{m} \right)$
$\Rightarrow \ \ mx^{2} + 8m^{2}x + 8 = 0\$has equal roots
$\Rightarrow \ 64\ m^{4} = 32m\ \ \lbrack\therefore D = 0\rbrack$
$\Rightarrow \ m = \frac{1}{\sqrt[3]{2}}$
$\therefore$ Equation of tangent is$\ y = \frac{1}{\sqrt[3]{2}}x + \sqrt[3]{2}$