Prove that $z - \frac{4}{z}$ is purely imaginary if and only if z is purely imaginary or |z|= 2. I am trying to prove that for any $z \neq 0$, $z - \frac{4}{z}$ is purely imaginary if and only if z is purely imaginary or |z|= 2. I tried writing = + and since z is purely imaginary, $x=0$ and $z - \frac{4}{z}$ = $iy - \frac{4}{iy}$. I don’t know if this makes sense. Any help is appreciated! Thank you!!

Let $z=x +i y$,

$$w=z-\frac4z = x+iy-\frac4{x+iy} = x+iy-\frac{4(x-iy)}{|z|^2}$$ $$= \left( 1-\frac 4{|z|^2}\right)x +i \left( 1+\frac 4{|z|^2}\right)y$$

Now, you could argue both ways.

$$x=0\>\>or \>\> |z|=2 \iff Re(w) =0$$