Question

Let \( z_1, z_2, z_3 \) be three complex numbers on the circle \( |z| = 1 \) with \( \arg(z_1) = -\frac{\pi}{4}, \arg(z_2) = 0 \) and \( \arg(z_3) = \frac{\pi}{4} \). If \( |z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}|^2 = \alpha + \beta \sqrt{2} \), where \( \alpha, \beta \in \mathbb{Z} \), then the value of \( \alpha^2 + \beta^2 \) is :

• A. 24
• B. 41
• C. 31
• D. 29

Hint:

For expressions involving complex numbers, use polar form and properties of modulus to simplify.

Answer 1

The Correct Option is D

Given \( z_1 = e^{-i\pi/4}, z_2 = 1, z_3 = e^{i\pi/4} \), we want to find the value of \( |z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}|^2 \). Substitute the values of \( z_1, z_2, z_3 \): \[ |z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}|^2 = \left| e^{-i\pi/4} \cdot 1 + 1 \cdot e^{i\pi/4} + e^{i\pi/4} \cdot e^{-i\pi/4} \right|^2 \] \[ = \left| 2e^{-i\pi/4} + i \right|^2 = \left| \sqrt{2} - \sqrt{2}i + i \right|^2 = 2 + 2 + 2\sqrt{2} = 29. \] Thus, \( \alpha^2 + \beta^2 = 29 \).