A curve in two dimensions is defined by the parameterised Cartesian coordinates
where the constants . Sketch the curve segment corresponding to the range . What is the length of the curve segment between the points and , as a function of ?
A geometrically sensitive ant walks along the curve with varying speed , where is the curvature at the point corresponding to parameter . Find the time taken by the ant to walk from to , where is a positive integer, and hence verify that this time is independent of .
[You may quote without proof the formula ]