If $Z_{1}, Z_{2}, Z_{3}$ are complex number such that $|z_{1}| = 2, |z_{2}| = 3, |z_{3}| = 4$ then maximum value of ${|z_{1} - z_{2}|}^{2} + {|z_{2} - z_{3}|}^{2} + {|z_{3} - z_{1}|}^{2}$ is

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Detailed Solution

$\begin{array}{l} {|z_{1} - z_{2}|}^{2} + {|z_{2} - z_{3}|}^{2} + {|z_{3} - z_{1}|}^{2} \\ = 2 ({|z_{1}|}^{2} + {|z_{2}|}^{2} + {|z_{3}|}^{2}) - (z_{1} {\bar{z}}_{2} + {\bar{z}}_{1} z_{2} + z_{2} {\bar{z}}_{3} + {\bar{z}}_{2} z_{3} + z_{3} {\bar{z}}_{1} + {\bar{z}}_{3} z_{1}) \\ = 58 - (z_{1} {\bar{z}}_{2} + {\bar{z}}_{1} z_{2} + z_{2} {\bar{z}}_{3} + {\bar{z}}_{2} z_{3} + z_{3} {\bar{z}}_{1} + {\bar{z}}_{3} z_{1}) \\ Now {|z_{1} + z_{2} + z_{3}|}^{2} \geq 0 \\ \Rightarrow {|z_{1}|}^{2} + {|z_{2}|}^{2} + {|z_{3}|}^{2} + z_{1} {\bar{z}}_{2} + {\bar{z}}_{1} z_{2} + z_{2} {\bar{z}}_{3} + {\bar{z}}_{2} z_{3} + z_{3} {\bar{z}}_{1} + {\bar{z}}_{3} z_{1} \geq 0 \\ \Rightarrow - (z_{1} {\bar{z}}_{2} + {\bar{z}}_{1} z_{2} + z_{2} {\bar{z}}_{3} + {\bar{z}}_{2} z_{3} + z_{3} {\bar{z}}_{1} + {\bar{z}}_{3} z_{1}) \leq 29 \end{array}$
from (1) and (2), maximum value = 58 + 29 = 87.