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Modulus of a Complex Number

If z1,z2,z3...

Question

If $z_{1}, z_{2}, z_{3}$ be three unimodular complex numbers then
$E = {| z_{1} - z_{2} |}^{2} + {| z_{2} - z_{3} |}^{2} + {| z_{3} - z_{1} |}^{2}$ 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 $9$
${| z_{1} - z_{2} |}^{2} = (z_{1} - z_{2}) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ (z_{1} - z_{2}) = (z_{1} - z_{2}) (_{1} - {¯ z}_{2})$
${| z_{1} |}^{2} + {| z_{2} |}^{2} - (z_{1} {¯ z}_{2} + {¯ z}_{1} z_{2})$
$= 1 + 1 - (z_{1} {¯ z}_{2} + ¯ ¯¯¯¯¯¯¯¯ ¯ z_{1} {¯ z}_{2}) = 2 - 2 R e (z_{1} {¯ z}_{2})$
$∴ E = (2 + 2 + 2) - 2 R e (z_{1} {¯ z}_{2} + z_{2} {¯ z}_{3} + z_{3} {¯ z}_{1})$ ...... $(1)$
Again ${| z_{1} + z_{2} + z_{3} |}^{2} \geq 0$
or $1 + 1 + 1 + 2 R e (z_{1} {¯ z}_{2} + z_{2} {¯ z}_{3} + z_{3} {¯ z}_{1}) \geq 0$
or $3 + (6 - E) \geq 0$ by $(1)$ $∴ E \leq 9$ .

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