Find the lengths of normal and subnormal at a point on the curve $y=\frac{a}{2}\left(e^{\frac{x}{a}}+e^{\frac{-x}{a}}\right)$

Given curve is $y=\frac{a}{2}\left(e^{x/a}+e^{-x/a}\right); y=a\cosh \left(\frac{x}{a}\right)$

diff w.r. to ‘$x$’

$m=\frac{dy}{dx}=a\sinh \left(\frac{x}{a}\right)\left(\frac{1}{a}\right)=\sin \mathrm{h}\left(\frac{x}{a}\right)$

Length of normal $=\left|y\sqrt{1+m^2}\right|$

$=\left|a\cos h\left(\frac{x}{a}\right)\sqrt{1+\sin h^2\left(\frac{x}{a}\right)}\right|$

$=\left|a\cos h\left(\frac{x}{a}\right)\sqrt{\cos h^2\left(\frac{x}{a}\right)}\right|=\left|a\cos h^2\frac{x}{a}\right|$

Length of sub normal

$=\left|ym\right|=\left|a\cosh \left(\frac{x}{a}\right)\sin h\left(\frac{x}{a}\right)\right|$