Complex Numbers
Equilateral triangle — sum of squared moduli
MJAT_TS6_P2
Grade 12
Question:
**Paragraph I:** $z_1^2+z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1$, $z_1+z_2+z_3=21$, $|z_1-z_2|=2\sqrt{3}$, $|z_1|=3\sqrt{3}$.
$|z_2|^2+|z_3|^2=$
Step-by-Step Solution
Key Concept: Equilateral $\Rightarrow|z_1|=|z_2|=|z_3|=3\sqrt{3}$? No — only $|z_1|=3\sqrt{3}$ given. Use $|z_1+z_2+z_3|^2=441$ and the constraint from equilateral condition.
$|z_2|^2+|z_3|^2=\mathbf{132}$.
Correct Answer: 132