Locus of the intersection of the tangents at the ends of the normal chords of the parabola $y^{2} = 4 a x$ is

(2 a + x) y^{2} + 4 a^{3} = 0

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(x + 2 a) y^{2} + 4 a^{2} = 0

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(y + 2 a) x^{2} + 4 a^{3} = 0

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None of these

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Solution

The correct option is A $(2 a + x) y^{2} + 4 a^{3} = 0$
Let $(h, k)$ be the point of intersection of tangents.
Then equation of chord of contact is $k y - 2 a (h + x) = 0$
$\Rightarrow 2 a x - k y + 2 a h = 0$ ....(1)
Equation of normal in parametric form: $y + x t - 2 a t - a t^{3} = 0$ ....(2)
Since, (1) and (2) are same
By comparing them $\frac{2 a}{t} = \frac{- k}{1} = \frac{2 a h}{- 2 a t - a t^{3}}$
By eliminating $t$ , we get $(2 a + h) k^{2} + 4 a^{3} = 0$

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