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Q. Number of distinct normals of a parabola passing through the focus of the parabola

Conic Sections

Solution:

Let the parabola be $y^2=4 a x$
Equation of any normal is
$y=-t x+2 a t+a t^3$
It passes through the focus $(a, 0)$
$0=-a t+2 a t+a t^3=a t\left(1+t^2\right)$
$\Rightarrow t =0$
$\therefore$ There is only 1 real normal.