Let $z$ be a complex number satisfying $|z^2-1|=|z{|}^2+1$. Then $z$ lies on

Solution

1. Given $|z^2-1|=|z{|}^2+1$, square both sides to obtain $|z^2-1{|}^2=(|z{|}^2+1{)}^2$.
2. Expand using the property $|w{|}^2=w\bar{w}$ to get $(z^2-1)({\bar{z}}^2-1)=(z\bar{z}+1{)}^2$.
3. Simplify the expression to $z^2{\bar{z}}^2-z^2-{\bar{z}}^2+1=z^2{\bar{z}}^2+2z\bar{z}+1$.
4. Subtract $z^2{\bar{z}}^2+1$ from both sides to yield $-(z^2+{\bar{z}}^2)=2|z{|}^2$.
5. Since $z^2+{\bar{z}}^2=2\text{Re}(z^2)$, the equation becomes $-2\text{Re}(z^2)=2|z{|}^2$, or $\text{Re}(z^2)+|z{|}^2=0$.
6. Let $z=x+iy$, then $x^2-y^2+x^2+y^2=0$, which simplifies to $2x^2=0$, implying $x=0$.
7. Hence the answer is (B).