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Question

If |z1|=|z2|=|z3|=|z4|=1 and z1+z2+z3+z4=0, then least value of the expressionE=|z1âˆ’z2|2+|z2âˆ’z3|2+|z3âˆ’z4|2+|z4âˆ’z1|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 8Given that, |z1|=|z2|=|z3|=|z4|=1and z1+z2+z3+z4=0 ....(1)⇒¯z1+¯z2+¯z3+¯z4=0 ....(2)Now, |z1−z2|2=(z1−z2)(¯z1−¯z2)=|z1|2+|z2|2−(z1¯z2+¯z1z2)E=|z1−z2|2+|z2−z3|2+|z3−z4|2+|z4−z1|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|2Since, |z1+z3|2≥0Therefore, E≥8Ans: B

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