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

To find the number of complex numbers \( z \), that satisfy \( |z| = 1 \) and

\(\left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \right| = 1,\)

we need to approach the problem using the properties of complex numbers and their magnitudes.

Given that \( |z| = 1 \), this implies \( z \) lies on the unit circle in the complex plane. So, if \( z = a + ib \), then \( a^2 + b^2 = 1 \).

The expression \( \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \) simplifies as follows:

Let \( z = e^{i\theta} \), thus \( \overline{z} = e^{-i\theta} \).

Then,

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

Therefore:

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

We need the absolute value of the above expression:

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

This simply happens when:

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

The solutions to \(\cos(2\theta) = \pm \frac{1}{2}\) are:

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

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

These angles represent 8 distinct solutions for \( z \), all having \( |z| = 1 \).

Therefore, the number of complex numbers \( z \) satisfying both conditions is 8.

Answer 2

Step 1: Understand the Condition

We are given that \( |z| = 1 \). This means that the complex number \( z \) lies on the unit circle in the complex plane.

Any complex number on the unit circle can be written as: \( z = e^{i\theta} \), where \( \theta \) is the argument (angle) of \( z \).

Step 2: Use the Given Equation

The equation provided is:

\[ \left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \right| = 1 \]

Step 3: Express \( z \) and \( \overline{z} \)

We know:

• \( z = e^{i\theta} \)
• \( \overline{z} = e^{-i\theta} \) (complex conjugate)

Now substitute into the equation:

\[ \frac{z}{\overline{z}} = \frac{e^{i\theta}}{e^{-i\theta}} = e^{2i\theta}, \quad \frac{\overline{z}}{z} = \frac{e^{-i\theta}}{e^{i\theta}} = e^{-2i\theta} \]

Step 4: Rewrite the Equation

Now the equation becomes:

\[ \left| e^{2i\theta} + e^{-2i\theta} \right| = 1 \]

Step 5: Use Euler’s Formula

We use the identity: \( e^{ix} + e^{-ix} = 2\cos x \)

Applying this, we get:

\[ e^{2i\theta} + e^{-2i\theta} = 2\cos(2\theta) \Rightarrow |2\cos(2\theta)| = 1 \]

Step 6: Solve for \( \cos(2\theta) \)

Divide both sides by 2:

\[ |\cos(2\theta)| = \frac{1}{2} \]

Step 7: Find All Possible \( 2\theta \) Values

The cosine of an angle is \( \pm \frac{1}{2} \) at the following values within \( 0 \leq 2\theta < 2\pi \):

• \( 2\theta = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} \)
• Also consider negative equivalents if needed: \( -\frac{\pi}{3}, -\frac{2\pi}{3}, -\frac{4\pi}{3}, -\frac{5\pi}{3} \)

So we have 8 distinct values of \( 2\theta \) that satisfy the condition.

Step 8: Find Corresponding \( \theta \) Values

Divide each \( 2\theta \) by 2 to get the angle \( \theta \):

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

Final Answer:

There are 8 distinct complex numbers \( z \) that satisfy the given condition.