Question

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

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

Answer 1

The correct option is B the imaginary axis

$|z^2-1|={\left|z\right|}^2+1$
let $z=x+iy$
$\Rightarrow |z^2-1|=\sqrt{(x^2-y^2-1{)}^2+(2xy{)}^2}$
but$|z^2-1|=x^2+y^2+1$
$\Rightarrow (x^2-y^2-1{)}^2+4x^2{y}^2=(x^2+y^2+1{)}^2$
$\Rightarrow 4x^2{y}^2=4x^2(y^2+1)$
$\Rightarrow x^2=0$
Hence, $z$ lies on the imaginary axis