Let z and \(\omega\) be two complex numbers such that \(\omega = z\overline{z} - 2z + 2\), \(\left|\frac{z+i}{z-3i}\right| = 1\) 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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Correct Answer: 4
Solution and Explanation
\(\omega = |z|^2 - 2z + 2 = (x^2 + 1) - 2x - 2i + 2 = (x^2 - 2x + 3) - 2i\).
\(\mathrm{Re}(\omega) = (x-1)^2 + 2\), minimum 2 at \(x = 1\). Then \(\omega = 2 - 2i = 2\sqrt2\, e^{-i\pi/4}\).
\(\omega^n\) is real when \(\frac{n\pi}{4}\) is an integer multiple of \(\pi\), i.e. n is a multiple of 4. Minimum n = 4 (\(\omega^4 = -64\)).
Answer: 4
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