21. Among the complex numbers $z$, find the numbers having the smallest and largest arguments, if $|z-10 i|=5 \sqrt{2}$.
21. (Fig. 21) Let the desired number be $\boldsymbol{z}=x+y i$.

Then $|x+y i-10 i|=5 \sqrt{2}$ or $x^{2}+(y-10)^{2}=50$ The locus of points (GMP), corresponding to the numbers $z$ and $10 i$, is the circle $x^{2}+(y-10)^{2}=50$. The smallest value $\bigodot_{1}=45^{\circ}, \quad z_{1}=55 i$. The largest value $\varphi_{2}=135^{\circ}, z_{2}=-5+5 i$.