A Reflected Chord's Best Area

The ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has eccentricity $\dfrac{\sqrt{3}}{2}$, and the quadrilateral formed by its foci and the endpoints of its minor axis has area $8\sqrt{3}$.

1. Find the equation of the ellipse $C$. 2. A line $l$ through the point $P(1, 0)$ meets $C$ at two points $A$ and $B$; the vertical line through $A$ meets $C$ again at $Q$. When the area of triangle $BPQ$ is largest, find the equation of $l$.