Explanation

(d)
The equation of any tangent to the parabola $y^{2} = 4x$ is
$y = mx + \frac{1}{m}\ \ \ \ \ \ \ldots(i)$
This touches the parabola $x^{2} = - 32y$, therefore the equation$\ x^{2} = - 32\ \left( mx + \frac{1}{m} \right)\$ has equal roots.
$\therefore(32\ m)^{2} = 4\ \left( \frac{32}{m} \right)\ \ \ \ \lbrack\therefore\ D^{2}\ = 4ac\rbrack$
$\Rightarrow \ 8m^{3} = 1\ \Rightarrow \ m = \frac{1}{2}$
On putting the value of $m$ is Eq. (i). we get
$x - 2y + 4 = 0$