Find $|z|$ if the given expression is purely imaginary

Question

Find $|z|$ if $\dfrac{z-2}{z+2}$ is entirely imaginary.

I know that if a number is purely imaginary, then $z-\overline{z}=2i$(some integer).

Answer 1

One may write $$ \frac{z-2}{z+2}=\frac{(z-2)(\bar{z}+2)}{(z+2)(\bar{z}+2)}=\frac{|z|^2-4+4i\:\Im z}{|z+2|^2} $$ and this is purely imaginery iff $|z|^2-4=0$.