Locus of the mid-points of the chords of the hyperbola  $x^2-y^2=a^2$ that touch the parabola $y^2=4ax$ is  

Let the  mid point  = (h, k) 
$\therefore$   equation of the chord   $xh-yk=h^2-k^2$
 $yk=xh+(k^2-h^2)$
 $y=\frac{xh}{k}+\frac{(k^2-h^2)}{k}$
 $\frac{k^2-h^2}{k}=\frac{ak}{h}$
$\Rightarrow k^{2}h-h^3=a{k}^2 \Rightarrow k^2(h-a)=h^3 \therefore x^3=y^2(x-a)$