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

The count of complex numbers \( z \) such that \( |z| = 1 \) and \( \left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \right| = 1 \) is determined using properties of complex numbers and their magnitudes.

The condition \( |z| = 1 \) indicates that \( z \) is on the unit circle. For \( z = a + ib \), this means \( a^2 + b^2 = 1 \).

The expression \( \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \) is simplified by setting \( z = e^{i\theta} \), which implies \( \overline{z} = e^{-i\theta} \).

This substitution yields:

\(\frac{z}{\overline{z}} = e^{2i\theta}\) and \(\frac{\overline{z}}{z} = e^{-2i\theta}\).

Thus,

\(\frac{z}{\overline{z}} + \frac{\overline{z}}{z} = e^{2i\theta} + e^{-2i\theta} = 2\cos(2\theta).\)

The requirement is:

\(|2\cos(2\theta)| = 1.\)

This equation simplifies to:

\(\cos(2\theta) = \pm \frac{1}{2}.\)

The solutions for \( 2\theta \) are:

\[ 2\theta = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{5\pi}{3}, \frac{4\pi}{3}, \frac{7\pi}{3}, \frac{8\pi}{3}, \cdots \]

Consequently, the values for \( \theta \) are:

\( \theta = \frac{\pi}{6}, \frac{\pi}{3}, \frac{5\pi}{6}, \frac{2\pi}{3}, \frac{7\pi}{6}, \frac{4\pi}{3}, \frac{11\pi}{6}, \frac{5\pi}{3} \).

These 8 distinct angles correspond to 8 unique solutions for \( z \) on the unit circle.

Therefore, there are 8 complex numbers \( z \) that satisfy both conditions.