Question

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

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

Answer 1

The correct option is A none of these

Let $z=x+iy$.
Then, $|z^2-1|=|z{|}^2+1$
$\Rightarrow |(x^2-y^2-1)+2ixy|=x^2+y^2+1$
$\Rightarrow (x^2-y^2-1{)}^2+4x^2{y}^2=(x^2+y^2+1{)}^2$
$\Rightarrow x^4+y^4+1-2x^2{y}^2+2y^2-2x^2+4x^2{y}^2$
$=x^4+y^4+1+2x^2{y}^2+2y^2+2x^2$
$\Rightarrow x=0$
Hence, $z$ lies on imaginary axis.
Hence, option (D) is the correct answer.