The locus of the mid points of the chords of the hyperbola

$\mathrm{T}=\mathrm{S}_{1}$

$\mathrm{xh}-\mathrm{yk}=\mathrm{h}^{2}-\mathrm{k}^{2}$

$\mathrm{y}=\frac{\mathrm{xh}}{\mathrm{k}}-\frac{\left(\mathrm{h}^{2}-\mathrm{k}^{2}\right)}{\mathrm{k}}$

this touches $y^{2}=8 x$ then $c=\frac{a}{m}$

$\left(\frac{\mathrm{k}^{2}-\mathrm{h}^{2}}{\mathrm{k}}\right)=\frac{2 \mathrm{k}}{\mathrm{h}}$

$2 y^{2}=x\left(y^{2}-x^{2}\right)$

$y^{2}(x-2)=x^{3}$