If z = x + i y , x y ? 0 , satisfies the equation z 2 + i z ¯ = 0 , then | z 2 | is equal to [2024]

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Solution

Q.
If $z = x + i y, x y \neq 0,$ satisfies the equation $z^{2} + i \bar{z} = 0,$ then $| z^{2} |$ is equal to [2024]

1 $9$

2 $\frac{1}{4}$

3 $1$

4 $4$

Ans.
(3)

We have, $z^{2} + i \bar{z} = 0$

$\Rightarrow z^{2} = - i \bar{z} \Rightarrow$ $| z^{2} |$ $=$ $| - i \bar{z} |$

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

$\Rightarrow | z |^{2} - | z | = 0$ $[∴ | \bar{z} | = | z |]$

$\Rightarrow$ $| z |$ $(| z | - 1) = 0$

$\Rightarrow | z | = 0 (not acceptable) or | z | = 1 \Rightarrow | z |^{2} = 1$

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