Question

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

• A. a circle
• B. a parabola
• C. an ellipse
• D. a straight line

Answer 1

The correct option is D a straight line

Let $z=x+iy.$ Then
$|z^2-1|=|z{|}^2+1\Rightarrow |(x^2-y^2-1)+2ixy|=x^2+y^2+1\Rightarrow (x^2-y^2-1{)}^2+4x^2{y}^2=(x^2+y^2+1{)}^2\Rightarrow (x^2-y^2-1{)}^2-(x^2+y^2+1{)}^2+4x^2{y}^2=0\Rightarrow \left(2x^2\right)(-2y^2-2)+4x^2{y}^2=0\Rightarrow -4x^2=0\therefore x=0$

Hence, $z$ lies on imaginary axis.