Points z in the complex plane satisfying `"Re"(z+1)^(2)=|z|^(2)+1` lie on

Options

A. a circle
B. a parabola
C. an ellipse
D. a hyperbola

Answer

B

Explanation

Let `z=x+iy`.
Then, `z+1=(x+1)+iy`
`rArr (z+1)^(2)=(x+1)^(2)-y^(2)+2i(x+1)y`
`rArr "Re"(z+1)^(2) = (x+1)^(2)-y^(2)`
Now, `"Re"(z+1)^(2)=|z|^(2)+1`
`rArr (x+1)^(2)-y^(2)=x^(2)+y^(2)+1`
`rArr x=y^(2)`, which is a parabola.