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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


E=2(|z1|2+|z2|2+|z3|2+|z4|2)
¯z1(z2+z4)¯z2(z1+z3)¯z3(z2+z4)¯z4(z1+z3)=2(|z1|2+|z2|2+|z3|2+|z4|2)
(¯z1+¯z3)(z2+z4)(¯z2+¯z4)(z1+z3)=8+|z1+z3|2+|z2+z4|28
z1+z3=(z2+z4)


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