Question

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

• A. A circle
• B. A parabola
• C. An ellipse
• D. None of these

Answer 1

Answer: (D)

None of these

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