The locus of the mid points of the chords of the hyperbola x ^ 2 -...

Equation of chord of hyperbola ${x}^{2} - {y}^{2} = 4$ with mid-point $\left(h, k\right)$ is given by $xh - yk = {h}^{2} - {k}^{2}$

$⇒ yk = xh - {h}^{2} + {k}^{2}$

$⇒ y = \dfrac{hx}{k} + \left(\dfrac{{k}^{2} - {h}^{2}}{k}\right)$

This will touch the parabola ${y}^{2} = 8x,$ if

$⇒ \dfrac{{k}^{2} - {h}^{2}}{k} = \dfrac{2}{h / k}$

$⇒ \dfrac{2k}{h} = \dfrac{{k}^{2} - {h}^{2}}{k}$

$⇒ {h}^{3} + 2{k}^{2} - {k}^{2}h = 0$
$∴$ Equation of locus is ${x}^{3} + 2{y}^{2} - {y}^{2}x = 0$

$⇒ {x}^{3} = {y}^{2}\left(x - 2\right)$