if `|z^2-1|=|z|^2+1` then `z` lies on

Options

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

Answer

D

Explanation

Let `z=z+iy`.Then,
`|z^(2)-1|=|z^(2)|+1`
`rArr |(x^(2)-y^(2)-1)+2ixy|=x^(2)+y^(2)+1`
`rArr (x^(2)-y^(2)-1)^(2)+4x^(2)y^(2)=(x^(2)+y^(2)+1)^(2)`
`rArr x =0`
Hence, z lies on imaginary axis.