 Solution

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.