Question

If the complex numbers \( z_1, z_2, 0 \) are vertices of an equilateral triangle, then \( z_1^2 + z_2^2 = \)

• A. \( 2z_1^2 z_2^2 \)
• B. \( z_1^2 z_2^2 \)
• C. \( 2z_1 z_2 \)
• D. \( z_1 z_2 \)

Hint:

Equilateral Triangle and Complex Numbers}
In the complex plane, equilateral triangles with one vertex at the origin satisfy the relation: \[ z_1^2 + z_2^2 = z_1 z_2. \]
Use vector rotation logic or geometric symmetry to derive such identities.

Answer 1

The Correct Option is D

If three complex numbers form an equilateral triangle with one vertex at the origin (0), and the others at \( z_1 \) and \( z_2 \), then the following identity holds: \[ z_1^2 + z_2^2 = z_1 z_2. \] This is a known result derived from geometrical symmetry in the complex plane and vector rotation by \( \pm 60^\circ \).