QJEEM21Ma8382: JEE Main 2021 Mathematics - Class 11 JEEM21Ma8382 Solution: Let z and be two complex numbers such that and Re( ) has…

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