The locus of the midpoint of the focal distance of a variable point moving on the parabola, y² = 4ax is a parabola whose

$\begin{array}{l}Given parabola y^2=4ax\\ focus \left(S\right)=(a,0)\\ Let P=(a{t}^2,2at) be any point on parabola\\ Let Q=(x,y) be locus of point\\ (\mathrm{x},\mathrm{y})=\left(\frac{\mathrm{a}+{\mathrm{at}}^2}{2},\mathrm{at}\right)\\ \Rightarrow 2x=\mathrm{a}+{\mathrm{at}}^2,y=at\\ \frac{2\mathrm{x}}{\mathrm{a}}=1+\frac{{\mathrm{y}}^2}{{\mathrm{a}}^2}\end{array}$
Parabola is $(\mathrm{y}-0{)}^2=2\mathrm{a}\left(\mathrm{x}-\frac{a}{2}\right)$