Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 10

Eliminating the parameter gives $x^2+y^2/4=1$, an ellipse centered at the origin with horizontal semiaxis one and vertical semiaxis two. At $t=0$ the particle is at $(1,0)$ with velocity $(0,2)$, so it travels counterclockwise, reaching $(0,2),(-1,0),(0,-2)$ successively and returning after $2\pi$. The right panel of the graph figure shows this direction.

For a parametric curve with $dx/dt\ne 0$, the chain rule givesThe specified point has $\cos t=\sin t=1/\sqrt{2}$, hence $t=\pi /4$ in the given traversal. Therefore