Question

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

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

Answer 1

Answer: (B)

the imaginary axis

Let $z=x+iy$
$z^2-1$$=(x^2-y^2-1)+2ixy$
$|z{|}^2+1$$=x^2+y^2+1$
Hence
$|z^2-1|=|z{|}^2+1$
$(x^2-y^2-1{)}^2+4x^2{y}^2=(x^2+y^2+1{)}^2$
$4x^2{y}^2=(x^2+y^2+1{)}^2-(x^2-y^2-1{)}^2$
$4x^2{y}^2=\left(2x^2\right)(2y^2+2)$
$x^2{y}^2=\left(x^2\right)(y^2+1)$
$x^2{y}^2=x^2{y}^2+x^2$
$x^2=0$
$x=0$
Hence, $z$ lies on imaginary axis.