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Q.

If $|z_{1}| = |z_{2}| = |z_{3}| = |z_{4}| = 1$ , and $z_{1} + z_{2} + z_{3} + z_{4} = 0$ then least value of the expression
$E = {|z_{1} - z_{2}|}^{2} + {|z_{2} - z_{3}|}^{2} + {|z_{3} - z_{4}|}^{2} + {|z_{4} - z_{1}|}^{2}$ is

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a

6

b

8

c

10

d

12

answer is B.

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Detailed Solution

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$\begin{array}{l} E = 2 ({|z_{1}|}^{2} + {|z_{2}|}^{2} + {|z_{3}|}^{2} + {|z_{4}|}^{2}) \\ - {\bar{z}}_{1} (z_{2} + z_{4}) - {\bar{z}}_{2} (z_{1} + z_{3}) \\ - {\bar{z}}_{3} (z_{2} + z_{4}) - {\bar{z}}_{4} (z_{1} + z_{3}) \\ = 8 + {|z_{1} + z_{3}|}^{2} + {|z_{2} + z_{4}|}^{2} \geq 8 \end{array}$

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