Question

If $|z^2-1|=|z{|}^2+1$, then z lies on

• A. the real axis
• B. an ellipse
• C. a circle
• D. the imaginary axis

Answer 1

The correct option is D the imaginary axis

Given that
$|z^2-1|=|z^2|+1\Rightarrow |z^2+(-1)|=|z^2|+|-1|$
It shows that the origin, -1, and $z^2$ lie on a line and $z^2$ and -1 lie on one side of the origin. Therefore, $z^2$ is a negative number. Hence, z will be purely imaginary. So we can say that z lies on the y-axis.