Thank you for reporting, we will resolve it shortly
Q.
The locus of the mid-point of the line segment joining the focus of the parabola $y ^{2}=4$ ax to a moving point of the parabola, is another parabola whose directrix is :
$h =\frac{ at ^{2}+ a }{2}, k =\frac{2 at +0}{2}$
$\Rightarrow t ^{2}=\frac{2 h - a }{ a }$ and $t =\frac{ k }{ a }$
$\Rightarrow \frac{ k ^{2}}{ a ^{2}}=\frac{2 h - a }{ a }$
$\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 $x-\frac{a}{2}=-\frac{a}{2} \Rightarrow x=0$
