Find the conditions for a co-normal point $(h, k)$ of a parabola $y^{2} = 4 a x$ .

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Solution

$P (h, k)$ for a parabola $y^{2} = 4 a x$ is a co-normal point.

We know that for a parabola $y^{2} = 4 a x$ normal is given by

$y = m x - 2 a m - a m^{3}$

$(h, k)$ will satisfy the equation.

$k = m h - 2 a m - a m^{3}$

$a m^{3} + m (2 a - h) + k = 0$

This is cubic in m, so it has three roots $m_{1}, m_{2}, m_{3}$ .

Therefore, $m_{1} + m_{2} + m_{3} = 0$ .....(1)
$m_{1} m_{2} + m_{2} m_{3} + m_{3} m_{1} = \frac{2 a - h}{a}$ .......(2)

$m_{1} m_{2} m_{3} = - \frac{k}{a}$ ....(3)

Therefore, equation 1, 2 and 3 are the three conditions.

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