21. Given complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1} \cdot z_{2}\right|=3, z_{1}+z_{2}=2 \mathrm{i}$, find the maximum and minimum values of $\left|z_{1}\right|$.
21. Let $z_{1}=r(\cos \theta+\mathrm{i} \sin \theta), z_{2}=\frac{3}{r}(\cos \varphi+\mathrm{i} \sin \varphi)$, from $z_{1}+z_{2}=2 \mathrm{i}$, we get
$$
\left\{\begin{array}{l}
r \cos \theta+\frac{3}{r} \cos \varphi=0, \\
r \sin \theta+\frac{3}{r} \sin \varphi=2 .
\end{array}\right.
$$
$\left(D^{2}+(2)^{2}\right)$, we get $r^{2}+\frac{9}{r^{2}}+6(\cos \theta \cdot \cos \varphi+\sin \theta \cdot \sin \varphi)=4$.
That is, $r^{2}+\frac{9}{r^{2}}+6 \cos (\theta-\varphi)=4$.
$$
\begin{array}{l}
\because r^{2}+\frac{9}{r^{2}}=4-6 \cos (\theta-\varphi) \leqslant 4+6=10, \\
\therefore 1 \leqslant r^{2} \leqslant 9, \therefore 1 \leqslant r \leqslant 3 .
\end{array}
$$

Therefore, the maximum value of $\left|z_{1}\right|$ is 3, and the minimum value is 1.