Let $|Z_{1}| = 1, |Z_{2}| = 2, |Z_{3}| = 3 Z_{1} + Z_{2} + Z_{3} = 3 + \sqrt{5} i$ , then the value of $Re (Z_{1} Z_{2} + Z_{2} Z_{3} + Z_{3} Z_{1})$ is equal to (where Z₁, Z₂ and Z₃ are complex numbers)

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$- \frac{1}{2}$

answer is D.

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Detailed Solution

$Z_{1} + Z_{2} + Z_{3} = 3 + \sqrt{5} i \Rightarrow {|Z_{1} + Z_{2} + Z_{3}|}^{2} = {|3 + \sqrt{5} i|}^{2}$
${|z_{1} + z_{2} + z_{3}|}^{2} = 14 {|z_{1}|}^{2} + {|z_{2}|}^{2} + {|z_{3}|}^{2} + 2 Re (z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1}) = 14 1 + 4 + 9 + 2 Re (z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1}) = 14 \Rightarrow 2 Re (z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1}) = 0$