Let z and w be two complex numbers such that w = z ¯ z - 2z + 2,...

$\omega = z\overset{¯}{z} - 2z + 2$

\left|\dfrac{z + i}{z - 3i}\right| = 1$</p><p>$⇒ \left|z + i\right| = \left|z - 3i\right|

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

$⇒ y = 1$

$⇒ z = x + i, x \in R$

$\omega = \left(x + i\right) \left(x - i\right)$ - 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(Re\left(\omega\right)\right), x = 1$</p><p>$⇒ \omega = 2 - 2i = 2 \left(1 - i\right) = 2\sqrt{2}{e}^{- i\dfrac{\pi}{4}}$</p><p>${\omega}^{n} = {\left(2\sqrt{2}\right)}^{n}{e}^{- i\dfrac{n\pi}{4}}$</p><p>For real and minimum value of$n$,</p><p>$n = 4