Question

Let $z$ and $w$ be two compex numbers such that $w=z\overline{z}-2z+2,\left|\frac{z+i}{z-3i}\right|=1$ and $Re\left(w\right)$ has minimum value. Then the minumum value of $n\in N$ for which $w^n$ is real, is equal to

Answer 1

Let $z=x+iy$
$|z+i|=|z-3i|\Rightarrow y=1$
Now
$w=x^2+y^2-2x-2iy+2\Rightarrow w=x^2+1-2x-2i+2$
$Re\left(w\right)=x^2-2x+3$
$=(x-1{)}^2+2$
$\therefore Re(w{)}_{\text{min}}\text{at}x=1\Rightarrow z=1+i$
Now $w=1+1-2-2i+2$
$w=2(1-i)=2\sqrt{2}{e}^{i(-\frac{\pi }{4})}{w}^n=2\sqrt{2}{e}^{i\left(\frac{-n\pi }{4}\right)}$
If $w^n$ is real
$\Rightarrow n=4$