"Let z and $\omega$ be two complex numbers such that $\omega = z\overline z - 2z + 2,\left| {{{z + i} \over {z - 3i}}} \right| = 1$ and Re($\omega$) has minimum value. Then, the minimum value of n $\in$ N for which $\omega$n is real, is equal to ______________."

Question

Accepted Answer

"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"

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

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

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