 Solution

For $ab\ne 0$, elimination of the trigonometric parameter givesAs real $\theta$ varies, the whole ellipse is traced, with semiaxes $|a|,|b|$. Signs of $a,b$ affect the traversal direction but not the set. If $a=0$, the locus is the vertical segment $x=0$, $|y|\le |b|$; if $b=0$, it is the horizontal segment $y=0$, $|x|\le |a|$. Both zero give only the origin.