12-16 If the complex number satisfies $z+|z|=2+8i$, then $|z|^{2}$ equals
(A) 68.
(B) 100.
(C) 169.
(D) 208.
(E) 289.
(39th American High School Mathematics Examination, 1988)
[Solution] Let $z=r(\cos \theta+i \sin \theta)$, then the given equation transforms to

Then
$$
r \cos \theta+i r \sin \theta+r=2+8 i,
$$

Thus $\left\{\begin{array}{l}r \cos \theta+r=2, \\ r \sin \theta=8,\end{array}\right.$
From (1) we have $r \cos \theta=2-r$,
(2) $)^{2}+(3)^{2}: \quad r^{2}=64+4-4 r+r^{2}$, which simplifies to $r=17$,
$$
\therefore \quad|z|^{2}=r^{2}=17^{2}=289 \text {. }
$$

Therefore, the answer is $(E)$.