Let $z$ is a complex number and $¯ ¯ ¯ z$ is conjugate of $z$ . If $(1 + i) z = (1 - i) ¯ ¯ ¯ z,$ then the value of $z + i ¯ ¯ ¯ z$ is

1

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Solution

The correct option is C $0$
Given : $(1 + i) z = (1 - i) ¯ ¯ ¯ z$
Now, let $z = a + i b$ , so
$z + i z = ¯ ¯ ¯ z - i ¯ ¯ ¯ z \Rightarrow z - ¯ ¯ ¯ z + i (z + ¯ ¯ ¯ z) = 0 \Rightarrow 2 i b + 2 i a = 0 \Rightarrow b = - a$
Therefore,
$z = a - i a = a (1 - i) ¯ ¯ ¯ z = a (1 + i) \Rightarrow z + i ¯ ¯ ¯ z = a [1 - i + i + i^{2}] ∴ z + i ¯ ¯ ¯ z = 0 (∵ i^{2} = - 1)$

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