Mathematics - Complex Number Question with Solution | TestHub

$z^4-|z{|}^4=4i{z}^2\Rightarrow z^4-{\left(z\overline{z}\right)}^2=4i{z}^2$

$\Rightarrow z^2\left(z^2-{\overline{z}}^2\right)=4i{z}^2 \Rightarrow z^2-{\overline{z}}^2=4i$

$\Rightarrow \left(z+\overline{z}\right)\left(z-\overline{z}\right)=4i$

$\Rightarrow \left(\frac{z+\overline{z}}{2}\right)\left(\frac{z-\overline{z}}{2i}\right)=1 \Rightarrow xy=1$

for $z_1 \& z_2 \Rightarrow x_1{y}_1=1$ and  $x_2{y}_2=1$

${\mathrm{x}}_1 \& {\mathrm{x}}_2$ are of opposite sign, similarly 

$y_1$ & $y_2$ are of opposite sign

$\Rightarrow x_1>0,y_1>0, x_2<0,y_2<0$

Now ${\left|z_1-z_2\right|}^2={\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2$

$=x_1^2+x_2^2+y_1^2+y_2^2-2x_1{x}_2-2y_1{y}_2$

$=x_1^2+x_2^2+y_1^2+y_2^2+x_1\left(-x_2\right)+x_1\left(-x_2\right)+y_1\left(-y_2\right)+y_1\left(-y_2\right)$

$\ge 8{\left(x_1^2·x_2^2·y_1^2·y_2^2·x_1\left(-x_2\right)·x_1\left(-x_2\right)·y_1\left(-y_2\right)·y_1\left(-y_2\right)\right)}^{1/8}$

$\ge 8{\left({\left(x_1{y}_1\right)}^4{\left(x_2{y}_2\right)}^4\right)}^{1/8}$

$\ge 8$.