The length of the normal to the curve $y=a\left(\frac{e^{-x / a}+e^{x / a}}{2}\right)$ at any point varies as the 

square of the ordinate of the point

We have,

$y=a\left(\frac{e^{-x / a}+e^{x / a}}{2}\right) \Rightarrow \frac{d y}{d x}=\frac{e^{x / a}-e^{-x / a}}{2}$

∴  Length of the normal at any point = $y \sqrt{1+\left(\frac{d y}{d x}\right)^2}$

$=y \sqrt{1+\left(\frac{e^{x / a}-a^{-x / a}}{2}\right)^2}=y\left(\frac{e^{x / a}+e^{-x / a}}{2}\right)=y\left(\frac{2 y}{a}\right)=\frac{2 y^2}{a}$

Hence, the length of the normal at any point varies as the square of the ordinate of the point.