The chord of the hyperbola x $^{2}$ - y $^{2}$ = a $^{2}$ touch the parabola y $^{2}$ = 4 ax , then the locus of the middle points of these chords in the curve.

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Solution

$\begin{matrix} L e t m i d - p o i n t s o f t h e c h o r d i s (h, k) ∴ E q u a t i o n o f c h o r d o f x^{2} - y^{2} = a^{2} i s T = S_{1} h x - k y = h^{2} - k^{2} ∴ y = \frac{h}{x} x - \frac{(h^{2} - k^{2})}{k} w h i c h i s tan g e n t o f y^{2} = 4 a x ∴ - \frac{(h^{2} - k^{2})}{k} = \frac{a}{\frac{h}{k}} o r, - (h^{2} - k^{2}) = \frac{a k^{2}}{h} o r, - h^{3} + h k^{2} = a k^{2} o r, k^{2} (h - a) = h^{3} H e n c e, l o c u s o f m i d - p o i n t s i s y^{2} (x - a) = x^{3} \end{matrix}$

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