An Angle Bisected by a Horizontal

The ellipse $C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) has eccentricity $\dfrac{\sqrt3}{2}$ and passes through $P(2, -1)$.

1. Find the equation of $C$. 2. Let $Q$ on $C$ be such that $PQ$ is horizontal. Two lines through $P$ meet $C$ at $A(x_1,y_1)$ and $B(x_2,y_2)$. If $PQ$ bisects $\angle APB$, prove that the slope of $AB$ is constant, and find it.