First slide

Algebra of complex numbers

Question

If z₁, ,z₂ , z₃ are complex numbers such that $|z_{1}| = |z_{2}| = |z_{3}| = |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}| = 1$ then $|z_{1} + z_{2} + z_{3}|$ is

Moderate

A

equal to 1
B

less than 1
C

greater than 3
D

equal to 3

Solution

$|z_{1}| = |z_{2}| = |z_{3}| = 1$
$\Rightarrow$ $z_{1} {\bar{z}}_{1} = z_{2} {\bar{z}}_{2} = z_{3} {\bar{z}}_{3} = 1$
$∴ |z_{1} + z_{2} + z_{3}| = |\frac{1}{{\bar{z}}_{1}} + \frac{1}{{\bar{z}}_{2}} + \frac{1}{{\bar{z}}_{3}}| = |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}| = 1$

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