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