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Question

If |z1|=|z2|=|z3|=|z4|=1 and z1+z2+z3+z4=0, then least value of the expression
E=|z1z2|2+|z2z3|2+|z3z4|2+|z4z1|2 is

A
6
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B
8
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C
10
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D
12
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Solution

The correct option is B 8
Given that, |z1|=|z2|=|z3|=|z4|=1
and 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)

E=|z1z2|2+|z2z3|2+|z3z4|2+|z4z1|2
E=2(|z1|2+|z2|2+|z3|2+|z4|2)(z1¯z2+¯z1z2)(z1¯z2+¯z1z2)(z1¯z2+¯z1z2)(z1¯z2+¯z1z2)
E=8[(z1+z3)(¯z2+¯z4)+(¯z1+¯z3)(z2+z4)]
E=8+[(z1+z3)(¯z1+¯z3)+(¯z1+¯z3)(z1+z3)] ....[ from (1) & (2)]
E=8+2|z1+z3|2
Since, |z1+z3|20
Therefore, E8
Ans: B

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