Question

If $|z^2-1|=\left|z^2\right|+1$, then $z$ lies on:

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

Answer 1

The correct option is B the imaginary axis

Let $z=x+iy$

$|z^2-1|=|z^2|+1=|z{|}^2+1=z\overline{z}+1$

$|(x+iy{)}^2-1|=(x+iy\left)\right(x-iy)+1$

$|x^2-y^2+2xyi-1|=x^2+y^2+1$

$\sqrt{(x^2-y^2-1{)}^2+4x^2{y}^2}=x^2+y^2+1$

Squaring both sides,

$x^4+y^4+1-2x^2{y}^2+2y^2-2x^2+4x^2{y}^2=x^4+y^4+1+2x^2{y}^2+2y^2+2x^2$

$4x^2=0⟹x=0$

Therefore $z$ lies on the imaginary axis.