If P, Q, R be three conormal points of the parabola $y^{2} = 4 a x$ whose normals pass through the point T, then show $S P . S Q . S R = a . S T^{2}$

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Solution

Let $P (a m_{1}^{2}, - 2 a m_{1})$ etc., then
$S P = a + x = a + a m_{1}^{2}$
$∴ S P . S Q . S R = a^{3} (1 + m_{1}^{2}) (1 + m_{2}^{2}) (1 + m_{3}^{2})$
$= a^{3} [1 + \sum m_{1}^{2} + \sum m_{1}^{2} m_{2}^{2} + m_{1}^{2} m_{2}^{2} m_{3}^{2}]$
$\sum m_{1}^{2} = {(\sum m_{1})}_{1}^{2} - 2 \sum m_{1} m_{2} = 0 - 2 (\frac{2 a - h}{a})$
$\sum m_{1}^{2} m_{2}^{2} = {(\sum m_{1} m_{2})}_{1}^{2} - 2 m_{1} m_{2} m_{3} \sum m_{1}$
$= {(\frac{2 a - h}{a})}^{2} - 0$
$m_{1}^{2} m_{2}^{2} m_{3}^{2} = \frac{k^{2}}{a^{2}}$
$∴$ SP.SQ.SR
$= a^{3} [1 - 2 (\frac{2 a - h}{a}) + {(\frac{2 a - h}{a^{2}})}^{2} + \frac{k^{2}}{a^{2}}]$
$= a [a^{2} - 2 a h + h^{2} + k^{2}] = a . S T^{2}$
where S is $(a, 0)$ and T is $(h, k)$ through which pass the three normals.

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