Question

Let \(z_1, z_2, z_3\) be three vertices of an equilateral triangle circumscribing the circle \(|z| = 1/2\). If \(z_1 = 1/2 + i\sqrt{3}/2\) and \(z_1, z_2, z_3\) were in anticlockwise sense, then \(z_2\) is

• A. \(1 + \sqrt{3}i\)
• B. \(1 - \sqrt{3}i\)
• C. 1
• D. -1

Hint:

For equilateral triangle, vertices are at 120° intervals.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
\(z_1 = e^{i\pi/3} = \cos 60° + i \sin 60° = 1/2 + i\sqrt{3}/2\) (magnitude 1, not 1/2).
Given circle \(|z| = 1/2\), vertices are at distance 1 from center? Actually equilateral triangle circumscribing circle means circle is inscribed.
From original: \(z_1 = 1/2 + i\sqrt{3}/2\), then \(z_2 = -1\).
Step 3: Final Answer:
\(z_2 = -1\).