Question

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

• A. a circle
• B. a parabola
• C. an ellipse
• D. none of these

Answer 1

The correct option is D none of these

Let
$z=x+iy$
$|z^2-1|=|z{|}^2+1$
$|(x^2-y^2-1)-i2xy|=x^2+y^2+1$
$(x^2-y^2-1{)}^2+4x^2{y}^2=(x^2+y^2+1{)}^2$
$4x^2{y}^2=(x^2+y^2+1{)}^2-(x^2-y^2-1{)}^2$
$4x^2{y}^2=\left(2x^2\right)(2y^2+2)$
$4x^2{y}^2=4x^2{y}^2+4x^2$
$4x^2=0$
$x=0$
Hence answer is none of these.