The locus of the mid-point of the line

$\mathrm{h}=\frac{\mathrm{at}^{2}+\mathrm{a}}{2}, \mathrm{k}=\frac{2 \mathrm{at}+0}{2}$

$\Rightarrow \mathrm{t}^{2}=\frac{2 \mathrm{~h}-\mathrm{a}}{\mathrm{a}}$ and $\mathrm{t}=\frac{\mathrm{k}}{\mathrm{a}}$

$\Rightarrow \frac{k^{2}}{a^{2}}=\frac{2 h-a}{a}$

$\Rightarrow$ Locus of $(h, k)$ is $y^{2}=a(2 x-a)$

$\Rightarrow y^{2}=2 a\left(x-\frac{a}{2}\right)$

Its directrix is $x-\frac{a}{2}=-\frac{a}{2} \Rightarrow x=0$