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Question

If z1+z2+z3+z4=0, then the expression |z1z2|2+|z2z3|2+|z3z4|2+|z4z1|22(|z1|2+|z2|2+|z3|2+|z4|2) is equal to 0, is and only if,

A
z1=z3 and z4=z2
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B
z1=z4 and z2=z3
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C
z1=z2 and z3=z4
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D
z1=z3 and z2=z4
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Solution

The correct option is A z1=z3 and z4=z2
Given that z1+z2+z3+z4=0 ....(1)
¯z1+¯z2+¯z3+¯z4=0 ....(2)

Now, |z1z2|2=(z1z2)(¯z1¯z2)=|z1|2+|z2|2(z1¯z2+¯z1z2)
|z1z2|2+|z2z3|2+|z3z4|2+|z4z1|22(|z1|2+|z2|2+|z3|2+|z4|2)=0
(z1¯z2+¯z1z2)(z1¯z2+¯z1z2)(z1¯z2+¯z1z2)(z1¯z2+¯z1z2)=0
(z1+z3)(¯z2+¯z4)+(¯z1+¯z3)(z2+z4)=0
(z1+z3)(¯z1+¯z3)(¯z1+¯z3)(z1+z3)=0 ....[ from (1) & from (2) ]
|z1+z3|2=0
z1=z3
Since, z1+z2+z3+z4=0
Therefore, z2=z4
Ans: A

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