Question

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

Answer 1

Answer: 4

$\omega =z\bar{z}-2z+2$
$\left|\frac{z+i}{z-3i}\right|=1$
$\Rightarrow |z+i|=|z-3i|$
$\Rightarrow z=x+i,x\in R$
$\omega =(x+i)(x-i)-2(x+i)+2$
$=x^2+1-2x-2i+2$
$\mathrm{Re}(\omega )=x^2-2x+3$
For min $(\mathrm{Re}(\omega )),x=1$
$\Rightarrow \omega =2-2i=2(1-i)=2\sqrt{2}{e}^{-i\frac{\pi }{4}}$
${\omega }^n=(2\sqrt{2}{)}^n{e}^{-1\frac{\pi \pi }{4}}$
For real & minimum value of $n$, $n=4$