12.33 If $|z|=1, \theta$ is the argument of $z$, then as $z$ varies, the image of $p=z^{2} +2 z \cos \theta$ in the complex plane is
(A) a circle.
(B) a hyperbola.
(C) a parabola.
(D) two intersecting lines.
(Chinese Jiangsu Province Suzhou City High School Mathematics Competition, 1990)
[Solution] Given $|z|=1$, let $z=\cos \theta+i \sin \theta(0 \leqslant \theta<2 \pi)$, then $p=z^{2}+2 z \cos \theta=(\cos 2 \theta+i \sin 2 \theta)+2\left(\cos ^{2} \theta+i \sin \theta \cos \theta\right)$ $=\left(\cos 2 \theta+2 \cos ^{2} \theta\right)+i(\sin 2 \theta+2 \sin \theta \cos \theta)$ $=(2 \cos 2 \theta+1)+i(2 \sin 2 \theta)$, which represents a circle with center at $(-1,0)$ and radius 2. Therefore, the answer is $(A)$.