Show that the length of the subnormal at any point on the curve $y^2=4ax$ is a constant.

Given curve is $y^2=4ax$

Diff w.r. to ‘$x$’

$2y\frac{dy}{dx}=4a$

$\Rightarrow \frac{dy}{dx}=\frac{4a}{2y}\Rightarrow m=\frac{dy}{dx}=\frac{2a}{y}$

Length of sub normal $=\left|ym\right|$

$=\left|y\frac{2a}{y}\right|=\left(2a\right)=\mathrm{constant}$