Chords of the hyperbola $x^{2} - y^{2} = a^{2}$ touch the parabola $y^{2} = 4 a x$ . Locus of their mid points is the curve

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

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(y - a) x^{3} = y^{4}

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(x - a) y^{4} = x^{4}

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

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Solution

The correct option is A $(x - a) y^{2} = x^{3}$
Let mid point of the chord is $p (h, k)$
Thus its equation is given by, $T = S_{1}$
$\Rightarrow h x - k y = h^{2} - k^{2} \Rightarrow y = \frac{h}{k} x + \frac{k^{2} - h^{2}}{k}$
Given this line is tangent to the parabola $y^{2} = 4 a x$
Thus Using condition of tangency, $c = \frac{a}{m}$
$\Rightarrow \frac{k^{2} - h^{2}}{k} = \frac{a k}{h} \Rightarrow (h - a) k^{2} = h^{3}$
Hence required locus is, $(x - a) y^{2} = x^{3}$

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