Question

If \(z_1, z_2\) and \(z_3\) are complex number such that \(|z_1| = |z_2| = |z_3| = \left|\frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3}\right| = 1\) then \(|z_1 + z_2 + z_3|\) is

• A. equal to 1
• B. less than 1
• C. greater than 3
• D. equal to 3

Hint:

If \(|z| = 1\), then \(1/z = \bar{z}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Since \(|z_k| = 1\), \(1/z_k = \bar{z}_k\).
Step 2: Detailed Explanation:
\(|z_k| = 1 \rightarrow 1/z_k = \bar{z}_k\)
Given \(\left|\frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3}\right| = |\bar{z}_1 + \bar{z}_2 + \bar{z}_3| = |z_1 + z_2 + z_3| = 1\)
Step 3: Final Answer:
\(|z_1 + z_2 + z_3| = 1\).