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Question

If z1,z2,z3 are three points lying on the circle |z|=2 then the minimum value of |z1+z2|2+|z2+z3|2+|z3+z1|2 is equal to

A
6
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B
12
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C
15
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D
24
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Solution

The correct option is B 12
|z1+z2|2+|z2+z3|2+|z3+z1|2
=2(|z1|2+|z2|2+|z3|2)+(z1¯z2+z2¯z1+z1¯z3+z3¯z1+z3¯z2+z2¯z3)
=24+(z1¯z2+z2¯z1+z1¯z3+z3¯z1+z3¯z2+z2¯z3)............1
Also we have, |z1+z2+z3|20
(z1¯z2+z2¯z1+z1¯z3+z3¯z1+z3¯z2+z2¯z3)12
Thus |z1+z2|2+|z2+z3|2+|z3+z1|212

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