Question

If complex numbers \(z_1,z_2\) and \(0\) are vertices of an equilateral triangle, then \(z_1^2+z_2^2-z_1z_2\) is equal to

• A. \(0\)
• B. \(z_1-z_2\)
• C. \(z_1+z_2\)
• D. \(1\)

Hint:

Use cube roots of unity for equilateral triangles.

Answer 1

The Correct Option is A

Step 1: For equilateral triangle with one vertex at origin: \[ z_2=z_1\omega \] where \(\omega^2+\omega+1=0\).
Step 2: \[ z_1^2+z_2^2-z_1z_2=z_1^2(1+\omega^2-\omega)=0 \]