Question

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

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

Answer 1

The correct option is B the imaginary axis

$|z^2-1|=|z{|}^2+1$

$|z^2-1{|}^2=(|z{|}^2+1{)}^2\Rightarrow (z^2-1\left)\right({\overline{z}}^2-1)=|z{|}^4+2|z{|}^2+1$

$\Rightarrow |z{|}^4-{\overline{z}}^2-z^2+1=|z{|}^4+2|z{|}^2+1$

$\Rightarrow {\overline{z}}^2+z^2+2|z{|}^2=0\Rightarrow {\overline{z}}^2+z^2+2z\overline{z}=0$

$\Rightarrow (z+\overline{z}{)}^2=0\Rightarrow z+\overline{z}=0$

$\Rightarrow z$ is purely imaginary. Hence $z$ lies on an imaginary axis.