A tangent to the parabola $x^{2} = 4 a y$ meets the hyperbola $x^{2} - y^{2} = a^{2}$ at two points P and Q, then midpoint of P and Q lies on the curve

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

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

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

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

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Solution

The correct option is B
$y^{3} = x^{2} (y - a)$

Equation of tangent to parabola $y = m x - a m^{2} . . . . . . (1)$

equation of chord of hyperbola whose midpoint is (h, k) is

$h x - k y = h^{2} - k^{2} . . . . . .$ (2)

form (1) and (2)

$\frac{m}{h} = \frac{1}{k} = \frac{a m^{2}}{h^{2} - k^{2}} \Rightarrow k^{3} = h^{2} (k - a)$

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