The locus of the mid-point of the chords of the hyperbola $x^2-y^2=4,$ that touches the parabola $y^2=8x$ is

Let midpoint of the chord be $P\left(h,k\right).$

Then, the equation of the chord will be $xh-yk=h^2-k^2$
$\Rightarrow y=\frac{h}{k}x+\frac{k^2-h^2}{k}$ … $\left(1\right)$
Now, the line $y=mx+c$ is tangent to $y^2=4ax$ if $c=\frac{a}{m}$
Hence, equation $\left(1\right)$ is tangent to $y^2=8x$ if
$\frac{k^2-h^2}{k}=\frac{2k}{h}\Rightarrow y^{2}x-x^3=2y^2$
$\Rightarrow y^2\left(x-2\right)=x^3$