A curve in two dimensions is defined by the parameterised Cartesian coordinates

$$x(u)=a e^{b u} \cos u, \quad y(u)=a e^{b u} \sin u$$

where the constants $a, b>0$. Sketch the curve segment corresponding to the range $0 \leqslant u \leqslant 3 \pi$. What is the length of the curve segment between the points $(x(0), y(0))$ and $(x(U), y(U))$, as a function of $U$ ?

A geometrically sensitive ant walks along the curve with varying speed $\kappa(u)^{-1}$, where $\kappa(u)$ is the curvature at the point corresponding to parameter $u$. Find the time taken by the ant to walk from $(x(2 n \pi), y(2 n \pi))$ to $(x(2(n+1) \pi), y(2(n+1) \pi))$, where $n$ is a positive integer, and hence verify that this time is independent of $n$.

[You may quote without proof the formula $\kappa(u)=\frac{\left|x^{\prime}(u) y^{\prime \prime}(u)-y^{\prime}(u) x^{\prime \prime}(u)\right|}{\left(\left(x^{\prime}(u)\right)^{2}+\left(y^{\prime}(u)\right)^{2}\right)^{3 / 2}} .$ ]