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 $z^4-|z{|}^4=4i{z}^2$, 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

Answer: 8

$z^4-|z{|}^4=4i{z}^2$
$z^4-z^2\cdot {\overline{z}}^2=4i{z}^2$
$\Rightarrow z^2=0$ or $z^2-{\overline{z}}^2=4i$
Let $z=x+iy$
$\Rightarrow (z+\overline{z})(z-\overline{z})=4i$
$\Rightarrow 2x\cdot 2iy=4i$
$\Rightarrow xy=1$
$z_1$ 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 $z_2=(-1,-1)$
$\Rightarrow {\left|z_1-z_2\right|}^2=8$