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Question

If z1,z2,z3 be three unimodular complex numbers then

E=|z1z2|2+|z2z3|2+|z3z1|2 cannot exceed


A
6
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B
9
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C
12
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D
none
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Solution

The correct option is D 9
|z1z2|2=(z1z2)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(z1z2)=(z1z2)(¯z1¯z2)
|z1|2+|z2|2(z1¯z2+¯z1z2)
=1+1(z1¯z2+¯¯¯¯¯¯¯¯¯¯z1¯z2)=22Re(z1¯z2)
E=(2+2+2)2Re(z1¯z2+z2¯z3+z3¯z1) ......(1)
Again |z1+z2+z3|20
or 1+1+1+2Re(z1¯z2+z2¯z3+z3¯z1)0
or 3+(6E)0 by (1) E9.

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