If $\left|z^2-1\right|=|z|^2+1$, then $z$ lies on

Question

Aicharya · Accepted Answer

$\left|z^2-1\right|^2=\left(|z|^2+1\right) \Rightarrow\left(z^2-1\right)\left(\bar{z}^2-1\right)=|z|^4+2|z|^2+1$ $\Rightarrow z^2+\bar{z}^2+2 z \bar{z}=0 \Rightarrow z+\bar{z}=0$ $\Rightarrow R(z)=0 \Rightarrow z$ lies on the imaginary axis.

If $\left|z^2-1\right|=|z|^2+1$, then $z$ lies on

Solution