The locus of the mid-point of the line segment joining the focus of...

$h = \dfrac{a{t}^{2} + a}{2}, k = \dfrac{2at + 0}{2}$

$⇒ {t}^{2} = \dfrac{2h - a}{a}$ and $t = \dfrac{k}{a}$

$⇒ \dfrac{{k}^{2}}{{a}^{2}} = \dfrac{2h - a}{a}$

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

$⇒ {y}^{2} = 2a\left(x - \dfrac{a}{2}\right)$

Its directrix is $x - \dfrac{a}{2} = - \dfrac{a}{2} ⇒ x = 0$