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Question

# If z1+z2+z3=0, then |z2âˆ’z3|2+|z3âˆ’z1|2+|z1âˆ’z2|2 equals

A
13|z1|2+2|z2|2+2|z3|2
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B
23(|z1|2+|z2|2+|z3|2)
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C
2(|z1|2+|z2|2+|z3|2)
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D
3(|z1|2+|z2|2+|z3|2)
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Solution

## The correct option is D 3(|z1|2+|z2|2+|z3|2)Given that, z1+z2+z3=0 ....(1)Let a=|z2−z3|2+|z3−z1|2+|z1−z2|2⇒a=(z2−z3)(¯z2−¯z3)+(z3−z1)(¯z3−¯z1)+(z1−z2)(¯z1−¯z2)⇒a=2(|z2|2+|z3|2+|z1|2)−[¯z2(z1+z3)+¯z3(z1+z2)+¯z1(z3+z2)]⇒a=2(|z2|2+|z3|2+|z1|2)+(¯z2z2+¯z3z3+¯z1z1) ....[ Using (1)]⇒a=3(|z2|2+|z3|2+|z1|2)Ans: D

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