Explanation

(c)
Let the coordinates of focus be $S\ (a,\ 0)$
Let any point on the parabola be $P\left( at^{2},\ 2at \right)$. Let the coordinates of mid point of $P$ and $S$ be $\left( x_{1},y_{1} \right)$
$\therefore\ \ x_{1} = \frac{a + at^{2}}{2},\ y_{1} = \frac{0 + 2at}{2}$
$\Rightarrow \ at^{2} = {2x}_{1} - a,\ \ y_{1} = at\$
$\Rightarrow \ a\left( \frac{y_{1}}{a} \right)^{2} = 2x_{1} - a$
$\Rightarrow \ y_{1}^{2}{= 2x}_{1}a - a^{2} =$Hence , the locus of the mid point is
$y^{2} = 2a\left( x - \frac{a}{2} \right)$
$\therefore$ Equation of directrix is $x - \frac{a}{2} = - \frac{a}{2}\ \Rightarrow x = 0$