Let $\mathrm{z}$ and $\mathrm{w}$ be two complex numbers such that
$\mathrm{w}=\mathrm{z} \overline{\mathrm{z}}-2 \mathrm{z}+2,\left|\frac{\mathrm{z}+\mathrm{i}}{\mathrm{z}-3 \mathrm{i}}\right|=1 \quad$ and $\quad \operatorname{Re}(\mathrm{w}) \quad$ has
minimum value. Then, the minimum value of $\mathrm{n} \in \mathbb{N}$ for which $\mathrm{w}^{\mathrm{n}}$ is real, is equal to_______.
$\omega=z \bar{z}-2 z+2$
$\left|\frac{z+i}{z-3 i}\right|=1$
$\Rightarrow|z+i|=|z-3 i|$
$\Rightarrow \quad z=x+i, \quad x \in \mathbb{R}$
$\omega=(x+i)(x-i)-2(x+i)+2$
$=x^{2}+1-2 x-2 i+2$
$\operatorname{Re}(\omega)=x^{2}-2 x+3$
For $\min (\operatorname{Re}(\omega)), x=1$
$\Rightarrow \quad \omega=2-2 \mathrm{i}=2(1-\mathrm{i})=2 \sqrt{2} \mathrm{e}^{-\mathrm{i} \frac{\pi}{4}}$
$\omega^{\mathrm{n}}=(2 \sqrt{2})^{\mathrm{n}} \mathrm{e}^{-\mathrm{i} \frac{\mathrm{n \pi}}{4}}$
For real \& minimum value of $\mathrm{n}$, $\mathrm{n}=4$
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