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
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
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For equilateral triangle, vertices are at 120° intervals.
Updated On: Apr 7, 2026
- \(1 + \sqrt{3}i\)
- \(1 - \sqrt{3}i\)
- 1
- -1
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The Correct Option is D
Solution and Explanation
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:
\(z_2 = -1\).
\(z_1 = e^{i\pi/3} = \cos 60° + i \sin 60° = 1/2 + i\sqrt{3}/2\) (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:
\(z_2 = -1\).
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