(2) For the parabola $y=a x^{2}+b x+1$, the parameters $a$ and $b$ satisfy $8 a^{2}+4 a b=b^{3}$. Then, as $a$ and $b$ vary, the vertex of the parabola must lie on ( ).
(A) Parabola
(B) Hyperbola
(C) Circle or Ellipse
(D) Line
(2) For the parabola $y=a x^{2}+b x+1$, the coordinates of the vertex are $\left(-\frac{b}{2 a}, \frac{4 a-b^{2}}{4 a}\right)$. Let $x=-\frac{b}{2 a}, y=\frac{4 a-b^{2}}{4 a}$, then we have $\frac{b}{a}=-2 x, y=1-\frac{b^{2}}{4 a}=1+\frac{b x}{2}$. Since $a \neq 0$, the conditions that $a$ and $b$ satisfy are equivalent to $8+4 \frac{b}{a}=b\left(\frac{b}{a}\right)^{2}$, thus we have
$$
8+4(-2 x)=b(-2 x)^{2}=4 x(2 y-2),
$$

which simplifies to $x y=1$. Therefore, the correct choice is B.