Question

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

• A. imaginary axis
• B. real axis
• C. line $x=y$
• D. line $x=2y$

Answer 1

The correct option is A imaginary axis

Let $z=x+iy$
$|z^2-1|=|z{|}^2+1$
$\Rightarrow |x^2-y^2+2xyi-1|=x^2+y^2+1$
$\Rightarrow (x^2-y^2-1{)}^2+4x^2{y}^2=(x^2+y^2+1{)}^2$
$\Rightarrow (x^2+y^2+1{)}^2-(x^2-y^2-1{)}^2=4x^2{y}^2\Rightarrow 2x^2(2y^2+2)=4x^2{y}^2\Rightarrow x=0$
$z$ lies on imaginary axis