Complex Numbers
Unit modulus complex numbers; integral values of |z₁+z₂+z₃|
MJMT_Full_Test_10
Grade 12
Question:
Let $z_1,z_2,z_3$ be three complex numbers such that $|z_1|=|z_2|=|z_3|=1$ and $\dfrac{z_1}{z_2z_3}+\dfrac{z_2}{z_3z_1}+\dfrac{z_3}{z_1z_2}=-1$. Number of possible integral values of $|z_1+z_2+z_3|$ is
Step-by-Step Solution
Key Concept: Given condition $\Rightarrow z_1^2+z_2^2+z_3^2=-z_1z_2z_3$. Combined with $|z_i|=1$: analyze $|z_1+z_2+z_3|^2=|z_1|^2+\ldots+2\text{Re}(\sum z_iz_j)$. Possible integral values are $\{1,2\}$ — count $=2$.
Number of integral values $=2$.
Correct Answer: 2