We have,
$$
\begin{aligned}
& \left|\frac{z-2 i}{z+2 i}\right|=2 \Rightarrow\left|\frac{x+i y-2 i}{\mid x+i y+2 i}\right|=2 \\
& \Rightarrow \quad \frac{\mid x+(y-2) i}{\mid x+(y+2) i} \mid=2 \Rightarrow \frac{|x+(y-2) i|}{|x+(y+2) i|}=2 \\
& \Rightarrow \frac{\sqrt{x^2+(y-2)^2}}{\sqrt{x^2+(y+2)^2}}=2 \\
& \Rightarrow \quad x^2+(y-2)^2=4\left[x^2+(y+2)^2\right] \\
& \Rightarrow \quad x^2+y^2-4 y+4=4 x^2+4 y^2+16 y+16 \\
& \Rightarrow 3 x^2+3 y^2+20 y+12=0 \\
& \Rightarrow \quad x^2+y^2+\frac{20}{3} y+4=0 \\
& \Rightarrow x^2+\left(y+\frac{10}{3}\right)^2+4-\frac{100}{9}=0 \\
& \Rightarrow \quad x^2+\left(y+\frac{10}{3}\right)^2=\left(\frac{8}{3}\right)^2 \\
&
\end{aligned}
$$
Which is equation of circle with centre $\left(0, \frac{-10}{3}\right)$ and radius $\frac{8}{3}$.