Question

If $|z^2-1|=|z{|}^2+1$, 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 $z=x+iy$.

Now,

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

or, $|x^2-y^2-1+i.2xy|=x^2+y^2+1$

or, $\sqrt{(x^2-y^2-1{)}^2+(2xy{)}^2}=x^2+y^2+1$

or, $(x^2-y^2-1{)}^2+4x^2{y}^2=(x^2+y^2+1{)}^2$

or, $(1+y^2+x^2{)}^2-(1+y^2-x^2{)}^2=4x^2{y}^2$

or, $4(1+y^2)\left(x^2\right)=4x^2{y}^2$

or, $4x^2=0$

or, $x=0$.

So $z$ lies on the imaginary axis.