First slide

Algebra of complex numbers

Question

If $|z_{1}| = |z_{3}| = |z_{3}| = 1$ then value of ${|z_{2} - z_{3}|}^{2} +$ ${|z_{3} - z_{1}|}^{2} + {|z_{1} - z_{2}|}^{2}$ cannot exceed

Moderate

A

3
B

6
C

9
D

12

Solution

$\begin{array}{l} |z_{1} + z_{2} + z_{3}| \geq 0 \\ \Rightarrow {|z_{1}|}^{2} + {|z_{2}|}^{2} + {|z_{3}|}^{2} + 2 Re (z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1}) \geq 0 \\ \Rightarrow Re (z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1}) \geq - \frac{3}{2} \end{array}$
Now, ${|z_{2} - z_{3}|}^{2} + {|z_{3} - z_{1}|}^{2} + {|z_{1} - z_{2}|}^{2}$
$= 6 - 2 Re (z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1})$
$\leq 6 - (- 3) = 9$

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