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# Find the conditions for a co-normal point (h,k) of a parabola $y^2=4ax$

$\begin{array}{1 1}(a)\;m_1+m_2+m_3=0\\(b)\;m_1m_2+m_2m_3+m_3m_1=\large\frac{2a-h}{a}\\(c)\;m_1m_2m_3=\large\frac{-k}{a}\\(d)\;\text{All of these}\end{array}$

P(h,k) for a parabola $y^2=4ax$ is a co-normal point.
We know that for a parabola $y^2=4ax$ normal is given by
$y=mx-2am-am^3$
(h,k) will satisfy the equation.
$k=mh-2am-am^3$
$am^3+m(2a-h)+k=0$
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)
$m_1m_2+m_2m_3+m_3m_1=\large\frac{2a-h}{a}$-------(2)
$m_1m_2m_3=\large\frac{-k}{a}$------(3)
equation(1),(2)&(3) are the three conditions.
Hence (d) is the correct answer.
edited Sep 23, 2014