Thank you for reporting, we will resolve it shortly
Q.
If chords of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4 a x$, then the locus of the middle points of these chords is the curve
Conic Sections
Solution:
Let mid point of the chord is $(h, k)$ equation of chord of $x^2-y^2=a^2$ is
$T = S , $
$\Rightarrow hx - ky = h ^2- k ^2$
$y =\frac{ h }{ k } x -\left(\frac{ h ^2- k ^2}{ k }\right)$.....(1)
(1) is tangent of $y^2=4 a x$
condition of tangencys $c =\frac{ a }{ m }$
$\Rightarrow-\frac{ h ^2- k ^2}{ k }=\frac{ a }{\left(\frac{ h }{ k }\right)} \Rightarrow- h ^2+ k ^2=\frac{ k ^2 a }{ h } $
$\Rightarrow- h ^3+ hk ^2= k ^2 a \Rightarrow k ^2( h - a )= h ^3$
Locus, $y^2(x-a)=x^3$