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Q.

The locus of middle points of normal chords of the parabola $y^{2} = 4 a x$ is

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a

$\frac{y^{2}}{2 a} + \frac{4 a^{3}}{y^{2}} = x - 2 a$

b

$\frac{y^{2}}{2 a} - \frac{4 a^{3}}{y^{2}} = x - 2 a$

c

$\frac{y^{2}}{2 a} + \frac{4 a^{3}}{y^{2}} = x + 2 a$

d

$\frac{y^{2}}{2 a} - \frac{4 a^{3}}{y^{2}} = x + 2 a$

answer is A.

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Detailed Solution

$\begin{array}{l} y^{2} = 4 a x \Rightarrow S_{1} = S_{11} \\ \Rightarrow y = \frac{2 a x}{y_{1}} + \frac{y_{1}^{2} - 2 a x_{1}}{y_{1}} \\ \Rightarrow c = - 2 a m - a m^{3} \Rightarrow \frac{y_{1}^{2} - 2 a x_{1}}{y_{1}} = - 2 a \frac{2 a}{y_{1}} - a {(\frac{2 a}{y_{1}})}^{3} \Rightarrow \frac{y^{2}}{2 a} + \frac{4 a^{3}}{y^{2}} = x - 2 a \end{array}$

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