Question

For a complex number $z$, let $Re(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^{4}-|z|^{4}=4 i z^{2}$, where $i=\sqrt{-1}$. Then the minimum possible value of $\left|z_{1}-z_{2}\right|^{2}$, where $z_{1}, z_{2} \in S$ with $Re\left( z _{1}\right)>0$ and $Re\left( z _{2}\right) < 0$, is ______.

Answer 1

$ z ^{4}-| z |^{4}=4 i z ^{2} $
$ z ^{4}- z ^{2} \cdot \overline{ z }^{2}=4 i z ^{2}$
$ \Rightarrow z ^{2}=0$ or $z ^{2}-\overline{ z }^{2}=4 i$
Let $ z = x + iy $
$ \Rightarrow ( z +\overline{ z })( z -\overline{ z })=4 i $
$ \Rightarrow 2 x \cdot 2 iy =4 i $
$\Rightarrow xy =1 $
$ z _{1} $ lies on $xy =1 $ in first quadrant
and $z _{2} $ lies on $ xy =1$ in third quadrant.
$ \Rightarrow \left| z _{1}- z _{2}\right|^{2} $ is minimum
when $ z _{1} \equiv(1,1) $ and $ z _{2}=(-1,-1)$
$ \Rightarrow \left| z _{1}- z _{2}\right|^{2}=8$