The GATE Grind

GATE 2026 CH – Question 40

Engineering Mathematics · Complex Variables: Complex numbers and polar form · 2 marks · Multiple choice

Consider the following complex numbers
$z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$
$z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$
where $r_1$, $r_2$ are real numbers, $0 \le \theta_1 \le \frac{\pi}{2}$, $0 \le \theta_2 \le \frac{\pi}{2}$, and $i = \sqrt{-1}$.

If $|z_1 + z_2| = |z_1| + |z_2|$, which one of the following conditions is necessarily CORRECT?

  1. $\theta_1 = 0, \theta_2 = \frac{\pi}{2}$
  2. $\theta_1 = \frac{\pi}{2}, \theta_2 = 0$
  3. $r_1 = r_2$
  4. $\theta_1 = \theta_2$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) $\theta_1 = \theta_2$

Explanation

Squaring both sides gives $r_1^2 + r_2^2 + 2r_1r_2\cos(\theta_1 - \theta_2) = r_1^2 + r_2^2 + 2r_1r_2$, so $\cos(\theta_1 - \theta_2) = 1$. With both angles between 0 and $\frac{\pi}{2}$ this means $\theta_1 = \theta_2$: the two numbers must point in the same direction. Equal moduli or the particular angles in A and B are not needed.