Two Meeting Points

Let $f(x) = \dfrac{1}{x}$ and $g(x) = ax^2 + bx$, where $a, b \in \mathbb{R}$ and $a \neq 0$. Suppose the graphs of $y = f(x)$ and $y = g(x)$ have exactly two common points $A(x_1, y_1)$ and $B(x_2, y_2)$. Determine, with proof, the signs of $x_1 + x_2$ and $y_1 + y_2$ in each of the cases $a < 0$ and $a > 0$.