If chords of the hyperbola $x^2-y^2=a^2$  touch the parabola $y^2=4ax$ , then the locus of the middle Points of these chords is the curve

The given hyperbola is $x^2-y^2=a^2$

Let $P\left(x_1,y_1\right)$ be any point on the required locus. 

The equation of the midpoint chord is $S_1=S_{11}$
$x{x}_1-y{y}_1={x_1}^2-{y_1}^2$

This can be written as 

$y=\frac{x_1}{y_1}x+\frac{{y_1}^2-{x_1}^2}{y_1}$

If the above line is tangent to the parabola $y^2=4ax$

Condition for tangency is $c=\frac{a}{m}$

$\begin{array}{rcl}\frac{{y_1}^2-{x_1}^2}{y_1}& =& \frac{a}{{\displaystyle \frac{x_1}{y_1}}}\\ {y_1}^{2}a& =& x_1\left({y_1}^2-{x_1}^2\right)\end{array}$

Therefore, the required locus is $\boxed{y^2\left(x-a\right)=x^3}$