The condition for the straight line $\mathrm{lx}+\mathrm{my}+\mathrm{n}=0$ to be a normal to the parabola ${\mathrm{y}}^2=4\mathrm{ax}\text{ is }{\mathrm{al}}^3+2{\mathrm{alm}}^2+{\mathrm{m}}^2\mathrm{n}=0$

$\mathrm{lx}+\mathrm{my}+\mathrm{n}=0 ... \left(1\right)$
is a normal to y² = 4ax ...(2)
Let P(t) be the foot of the normal, equation of the normal is $\mathrm{y}+\mathrm{xt}=2\mathrm{at}+{\mathrm{at}}^3$
$\Rightarrow \mathrm{xt}+\mathrm{y}-\left(2\mathrm{at}+{\mathrm{at}}^3\right)=0 ...\left(3\right)$
(1) & (3) represent the same line
$\begin{array}{l}\Rightarrow \frac{\mathrm{l}}{\mathrm{t}}=\frac{\mathrm{m}}{1}=\frac{\mathrm{n}}{-\left(2\mathrm{at}+{\mathrm{at}}^3\right)}\\ \Rightarrow \mathrm{t}=\frac{\mathrm{l}}{\mathrm{m}}\text{ and }2\mathrm{at}+{\mathrm{at}}^3=-\frac{\mathrm{n}}{\mathrm{m}}\\ \Rightarrow 2\mathrm{a}\left(\frac{\mathrm{l}}{\mathrm{m}}\right)+\mathrm{a}\frac{{\mathrm{l}}^3}{{\mathrm{m}}^3}=\frac{-\mathrm{n}}{\mathrm{m}}\\ \Rightarrow {\mathrm{al}}^3+2{\mathrm{alm}}^2+{\mathrm{m}}^2\mathrm{n}=0\end{array}$