Complex Numbers
Algebra of Complex Numbers
Grade Class 11
Question:
<p>Let \( z_1, z_2, z_3 \) be distinct complex numbers with \( |z_1|=|z_2|=|z_3|=a \) and centroid at the origin. Then \( |z_1 z_2 + z_2 z_3 + z_3 z_1| \) equals:</p>
Step-by-Step Solution
Key Concept: Centroid at origin: z_1+z_2+z_3 = 0. Then (z_1+z_2+z_3)^2 = 0 \Rightarrow z_1^2+z_2^2+z_3^2+2(z_1z_2+z_2z_3+z_3z_1) = 0. Also use z_1z_2z_3 = a^2(z_1+z_2+z_3) / something.
<p>$ z_1+z_2+z_3=0 \Rightarrow (z_1+z_2+z_3)^2=0 \Rightarrow \sum z_k^2 = -2\sum_{cyc} z_1 z_2 $. Since $ |z_k|=a $, $ z_k \bar{z}_k = a^2 $. Taking conjugates and using $ \bar{z}_k = a^2/z_k $: $ \sum 1/z_k = 0 \Rightarrow z_1 z_2+z_2 z_3+z_3 z_1 = 0 \cdot z_1 z_2 z_3 / ... $. Final: $ |z_1 z_2+z_2 z_3+z_3 z_1| = a^2 $.</p>
Correct Answer: B