First slide

Algebra of complex numbers

Question

If $z_{1}, z_{2}, z_{3}$ be three complex number such than $z_{1} + z_{2} + z_{3} = 0$ and $|z_{1}| = |z_{2}| = |z_{3}| = 1$ than $|z_{1}^{2} + 2 z_{2}^{2} + z_{3}^{2}|$ equals

Moderate

A

1
B

2
C

3
D

4

Solution

$\begin{array}{l} z_{1} + z_{2} + z_{3} = 0 \Rightarrow {\bar{z}}_{1} + {\bar{z}}_{2} + {\bar{z}}_{3} = 0 \\ \Rightarrow \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} = 0 \end{array} [∵ z_{1} {\bar{z}}_{1} = 1 etc.]$
Now, $0 = {(z_{1} + z_{2} + z_{3})}^{2} = z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + 2 (z_{2} z_{3} + z_{3} z_{1} + z_{1} z_{2})$
$\begin{array}{l} \Rightarrow 0 = z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + 2 z_{1} z_{2} z_{3} (\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}) \\ = z_{1}^{2} + z_{2}^{2} + z_{3}^{2} \end{array}$
Thus, $|z_{1}^{2} + 2 z_{2}^{2} + z_{3}^{2}| = |z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{2}^{2}|$
$= |0 + z_{2}^{2}| = {|z_{2}|}^{2} = 1$

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