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

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

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