SAT1000 - P941

This is known as a Tusi couple , and the following animation (courtesy of Wikipedia) shows the locus of points for $N$ :

By symmetry, $M$ will be the locus of points perpendicular to the locus of points for $N$ , making the answer $\boxed{A}$ .

Though wanted to post animation, which is best to answer this problem, but I am yet to learn the skill. I am showing the theoretical proof here if you are interested.

Since the big circle has twice the radius of the small one, its circumference is also twice that of the small one. This means that for the small circle to go one round the big one, itself has rotated two cycles or $4\pi$ . This means that the center $P$ of the small circle moves one round. points $M$ and $N$ on the circumference would move two rounds. This also means that when $P$ moves through an angle $\theta$ counterclockwise, $M$ and $N$ move through $2\theta$ . As $\triangle OPM$ is always an isosceles triangle with equal sides $r = \frac 12$ , $\angle OMP = \angle MOP = \theta$ . And the external $\angle MPQ$ is always $2\theta$ , $M$ is always on the $x$ -axis. Similarly, $N$ is always on the $y$ -axis.