Let z and w be two complex numbers such that w = z z ¯ - 2 z + 2 , z + i z - 3 i = 1 and Re w has…

$\omega =z\overline{z}-2z+2$

$\left|\frac{z+i}{z-3i}\right|=1$

$\Rightarrow \left|z+i\right|=\left|z-3i\right|$

$\Rightarrow x^2+{\left(y+1\right)}^2=x^2+{\left(y-3\right)}^2$

$\Rightarrow y=1$

$\Rightarrow z=x+i, x\in R$

$\omega =\left(x+i\right)\left(x-i\right)-2\left(x+i\right)+2$

$=x^2+1-2x-2i+2$

$Re\left(\omega \right)=x^2-2x+3={\left(x-1\right)}^2+2$

For min $\left(\mathrm{Re}\left(\omega \right)\right), x=1$

$\Rightarrow \omega =2-2i=2\left(1-i\right)=2\sqrt{2}{e}^{-i\frac{\pi }{4}}$

${\omega }^n={\left(2\sqrt{2}\right)}^n{e}^{-i\frac{n\pi }{4}}$

For real and minimum value of $n$,

$n=4$