In the curve $y = c{e}^{x/a}$ the

We have $\mathrm{y}={\mathrm{ce}}^{\mathrm{xla}}\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{c}}{\mathrm{a}}{\mathrm{e}}^{\mathrm{xla}}=\frac{1}{\mathrm{a}}\mathrm{y}$
Or $\frac{\mathrm{y}}{\mathrm{dy}/\mathrm{dx}}=$a constant or sub tangent = constant
Or Length of the sub-normal $=\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}\frac{\mathrm{y}}{\mathrm{a}}=\frac{{\mathrm{y}}^2}{\mathrm{a}}$
$\infty$ (Squar of the ordinate) 
Equation of the tangent at
$\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)\text{ is }\mathrm{y}-{\mathrm{y}}_1=\frac{{\mathrm{y}}_1}{\mathrm{a}}\left(\mathrm{x}-{\mathrm{x}}_1\right)\text{. }$
This meets the x-axis at a point given by
$\mathrm{y}-{\mathrm{y}}_1=\frac{{\mathrm{y}}_1}{\mathrm{a}}\left(\mathrm{x}-{\mathrm{x}}_1\right)\text{ or }\mathrm{x}={\mathrm{x}}_1-\mathrm{a}$
The curve meets the y-axis at (0,c) Therefore
$\begin{array}{l}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{(0,\mathrm{c})}=\frac{\mathrm{c}}{\mathrm{a}}\\ \Rightarrow \mathrm{y}-\mathrm{c}=-\frac{1}{\mathrm{c}/\mathrm{a}}(\mathrm{x}-0)\text{ or }\mathrm{ax}+\mathrm{cy}={\mathrm{c}}^2\end{array}$