Given, parabola, $y^{2}=4 ax$
$\Rightarrow 2y \frac{dy}{dx}=4a$
$\Rightarrow \frac{dy}{d x}=\frac{2a}{y} \, \dots(i)$
which is the slope of tangent.
Given, $lx+my+n=0$
is an equation of tangent of the parabola
$y^{2}=4 a x$
$\therefore $ Slope of tangent $=-\frac{l}{m}\, \dots(ii)$
From Eqs. (i) and (ii)
$\frac{2a}{y}=-\frac{l}{m} $
$\Rightarrow y=\frac{-2 a m}{l}$
$\because y^{2}=4 a x$
$ \Rightarrow \frac{4 a^{2} m^{2}}{l^{2}}=4 a x$
$\Rightarrow x=\frac{am^{2}}{l^{2}}$
On putting the values of x and y in the following equation
$lx+my+n=0$
$l\left(\frac{a m^{2}}{l^{2}}\right)+m\left(\frac{-2 a m}{l}\right)+n=0$
$\frac{a m^{2}}{l}-\frac{2 a m^{2}}{l}+n=0$
$\Rightarrow \frac{a m^{2}}{l}=n$
$ \Rightarrow a m^{2}=nl$
which is the required relation