JEE Main · MathematicsMedium

$If the complex numbers z_{1}, z_{2}, and z_{3} represent the vertices of an equilateral triangle, then z_{1}^{2} + z_{2}^{2} + z_{3}^{2} equals$

A. $z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}$
B. $0$
C. $z_{1} + z_{2} + z_{3}$
D. $2 (z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1})$

Show correct answer & step-by-step solution

Correct answer: A — $z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}$

Solution

For an equilateral triangle, the condition is z_{1}^{2} + z_{2}^{2} + z_{3}^{2} - z_{1} z_{2} - z_{2} z_{3} - z_{3} z_{1} = 0 . Rearranging this gives the required identity.

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If the complex numbers z1, z2, and z3 represent the vertices of an equilateral t… — JEE Main Mathematics Question with Solution |