If $| z | = max {| z - 1 |, | z + 1 |}$ , then

∣ ∣ z + ¯ ¯ ¯ z ∣ ∣ = \frac{1}{2}

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z + ¯ ¯ ¯ z = 1

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∣ ∣ z + ¯ ¯ ¯ z ∣ ∣ = 1

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Solution

The correct option is B $∣ ∣ z + ¯ ¯ ¯ z ∣ ∣ = 1$
We have, $| z | = | z - 1 |$

$\Rightarrow {| z |}^{2} = {| z - 1 |}^{2}$

$\Rightarrow z ¯ ¯ ¯ z = (z - 1) (¯ ¯ ¯ z - 1)$

$\Rightarrow z ¯ ¯ ¯ z = z ¯ ¯ ¯ z - z - ¯ ¯ ¯ z + 1 \Rightarrow z + ¯ ¯ ¯ z = 1$

Also, if $| z | = | z + 1 | \Rightarrow {| z |}^{2} = {| z + 1 |}^{2}$

$\Rightarrow z ¯ ¯ ¯ z = (z + 1) (¯ ¯ ¯ z + 1) = z ¯ ¯ ¯ z + z + ¯ ¯ ¯ z + 1$

$\Rightarrow z + ¯ ¯ ¯ z = - 1$

$∴ ∣ ∣ z + ¯ ¯ ¯ z ∣ ∣ = 1$

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