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If $\left|z^2-1\right|=|z|^2+1$, then $z$ lies on
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Verified Answer
The correct answer is:
the imaginary axis.
the imaginary axis.
$\left|z^2-1\right|^2=\left(|z|^2+1\right) \Rightarrow\left(z^2-1\right)\left(\bar{z}^2-1\right)=|z|^4+2|z|^2+1$
$\Rightarrow z^2+\bar{z}^2+2 z \bar{z}=0 \Rightarrow z+\bar{z}=0$
$\Rightarrow R(z)=0 \Rightarrow z$ lies on the imaginary axis.
$\Rightarrow z^2+\bar{z}^2+2 z \bar{z}=0 \Rightarrow z+\bar{z}=0$
$\Rightarrow R(z)=0 \Rightarrow z$ lies on the imaginary axis.
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