[tex]\displaystyle Fie~z=x+yi,~cu~z,y \in \mathbb{R}. \\ \\ |z|= \sqrt{x^2+y^2},~iar~ecuatia~devine: \\ \\ \sqrt{x^2+y^2}-2x-2yi+2i=0 \Leftrightarrow \\ \\ \sqrt{x^2+y^2}-2x+(2-2y)i=0 . \\ \\ Deci~ \left \{ {{\sqrt{x^2+y^2}-2x=0} \atop {2-2y=0}} \right. . \\ \\ 2-2y=0 \Rightarrow \boxed{y=1}~. \\ \\ \sqrt{x^2+y^2}-2x=0 \Leftrightarrow \sqrt{x^2+1}=2x \Rightarrow x \geq 0~si~x^2+1=4x^2 \Rightarrow \\ \\ \Rightarrow \boxed{x= \frac{\sqrt{3}}{3}}~.[/tex]
[tex]SOLUTIE:~z= \frac{ \sqrt{3}}{3}+i.[/tex]