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8. If tangents be drawn from points on the line $x = c$ to the parabola $y^2 = 4ax$, shew that the locus of the intersection of the corresponding normals is the parabola
$$ay^2 = c^2(x+c-2a).$$

Solution:

$\;\;\;$ The intersection of the tangents at $m$ and $m'$ is the point $amm', a(m + m'),$ and the intersection of the normals is $a (m^2 + mm' + m'^2) + 2a,\;\; -amm'(m + m')$.

Now $amm' = c$. Hence, putting $m + m' = \lambda,$ the latter point is
$\;\;\;\displaystyle x = a\left(\lambda^2 - \frac{c}{a}\right) + 2a,\;\; y = -c\lambda.$
Eliminating $\lambda,$ the result follows.