Radius of curvature at given point

Question

The value of radius of curvature for the curve $$s = ce^{x/c}$$ at $(s,\psi)$

I know that radius of curvature is $$ \frac{(1+y_1)^2}{y_2}$$ how to apply these formula here

Answer 1

Given $y=c e^{x/c}$ the curvature in cartesian coordinates is $$\kappa(x)=\frac {|y''|}{\left(1+{y'}^{2}\right)^{\frac {3}{2}}}$$ So in this case $$\kappa(x)=\frac{e^{x/c}}{\left| c\right| \left(e^{\frac{2 x}{c}}+1\right)^{3/2}}$$ radius of curvature is $r(x)=\frac{1}{\kappa(x)}$.

In this case $$r(x)=\left| c\right| e^{-\frac{x}{c}} \left(e^{\frac{2 x}{c}}+1\right)^{3/2}$$