Question

The number of complex numbers \( z \), satisfying \( |z| = 1 \) and \[ \left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \right| = 1, \] is:

• A. 6
• B. 4
• C. 10
• D. 8

Hint:

When solving problems involving complex numbers on the unit circle, convert to polar form, use trigonometric identities, and check for distinct solutions in the given interval.

Answer 1

The Correct Option is D

We are given the conditions: \[ |z| = 1 \quad \text{and} \quad \left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \right| = 1. \] Since \( |z| = 1 \), we know that \( z \) lies on the unit circle, so we can write: \[ z = e^{i\theta}, \quad \overline{z} = e^{-i\theta}. \] Substituting into the second condition, we get: \[ \left| e^{2i\theta} + e^{-2i\theta} \right| = 1. \] Using the identity \( e^{ix} + e^{-ix} = 2\cos(x) \), we get: \[ \left| 2\cos(2\theta) \right| = 1 \quad \Rightarrow \quad |\cos(2\theta)| = \frac{1}{2}. \] The solutions to \( |\cos(2\theta)| = \frac{1}{2} \) occur at: \[ 2\theta = \pm \frac{\pi}{3} + 2k\pi, \quad k \in {Z}. \] Thus, the solutions for \( \theta \) are: \[ \theta = \pm \frac{\pi}{6} + k\pi. \] The distinct values of \( \theta \) in the interval \( [0, 2\pi) \) are: \[ \theta = \frac{\pi}{6}, \frac{\pi}{3}, \frac{5\pi}{6}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}. \] Thus, there are 6 distinct solutions for \( \theta \), corresponding to 6 distinct complex numbers.