The locus of the mid points of chords of the parabola $y^{2} = 8 x,$ which subtends a right angle at its vertex, is another parabola whose length of latus rectum is equal to

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Solution

Let $P (h, k)$ be the mid point of chord $Q R$ of the parabola $y^{2} = 8 x$ .
Equation of $Q R$ is $T = S_{1}$
$\Rightarrow y k - 4 (x + h) = k^{2} - 8 h$
$\Rightarrow y k - 4 x = k^{2} - 4 h$

Let $A (0, 0)$ be the vertex of the parabola.
Given, $A Q$ is perpendicular to $A R$
$\Rightarrow$ Combined equation of $A Q$ and $A R$ is : $y^{2} - 8 x (\frac{y k - 4 x}{k^{2} - 4 h}) = 0$ (by homogenization)
For mutually perpendicular pair of straight lines,
coefficient of $x^{2}$ $+$ coefficient of $y^{2} = 0$
$\Rightarrow \frac{32}{k^{2} - 4 h} + 1 = 0$
$\Rightarrow k^{2} - 4 h + 32 = 0$
$∴$ Locus of $P (h, k)$ is $y^{2} - 4 x + 32 = 0$
$\Rightarrow y^{2} = 4 (x - 8)$
Length of latus rectum $= 4$

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