If the 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

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

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

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

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Solution

The correct option is B $y^{2} (x - a) = x^{3}$
Equation of chord of hyperbola $x^{2} - y^{2} = a^{2}$ with mid-point as $(h, k)$ is given by,
$x h - y k = h^{2} - k^{2}$
$\Rightarrow y = \frac{h}{k} x - \frac{(h^{2} - k^{2})}{k}$
This will touch the parabola $y^{2} = 4 a x$ , if
$- (\frac{h^{2} - k^{2}}{k}) = \frac{a}{\frac{h}{k}}$
$\Rightarrow a k^{2} = h^{3} + k^{2} h$
Thus locus of the mid-point is $x^{3} = y^{2} (x - a)$ .

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