Let z = x + iy
| z + i | = | z $-$ 3i |
$\Rightarrow$ y = 1
Now
$\omega$ = x² + y² $-$ 2x $-$ 2iy + 2
$\omega$ = x² + 1 $-$ 2x $-$ 2i + 2
Re($\omega$) = x² $-$ 2x + 3
Re($\omega$) = (x $-$ 1)² + 2
Re($\omega$)_min at x = 1 $\Rightarrow$ z = 1 + i
Now,
$\omega$ = 1 + 1 $-$ 2 $-$ 2i + 2
$\omega$ = 2(1 $-$ i) = 2$\sqrt 2 {e^{i\left( {{{ - \pi } \over 4}} \right)}}$
$\omega$ⁿ = 2$\sqrt 2 {e^{i\left( {{{ - n\pi } \over 4}} \right)}}$
If $\omega$ⁿ is real $\Rightarrow$ n = 4