Complex Numbers
Modulus and Conjugate Properties
Premium Question
Grade 11
Question:
If $z_1 + z_2 + z_3 = 0$ and $|z_1| = |z_2| = |z_3| = 1$, then $z_1z_2 + z_2z_3 + z_3z_1$ equals:
(1) $1$
(2) $-1$
(3) $3$
(4) $0$
Step-by-Step Solution
Key Concept: Since $|z_k| = 1$ we have $z_k^* = 1/z_k$. Compute $(z_1z_2 + z_2z_3 + z_3z_1)^*$ using these reciprocals and recall that $z_1 + z_2 + z_3 = 0$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (4)