A Scaled-Ellipse Area Maximum

In the coordinate plane $xOy$, the ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has eccentricity $\dfrac{\sqrt 3}2$ and left and right foci $F_1$, $F_2$. The circle of radius $3$ centered at $F_1$ and the circle of radius $1$ centered at $F_2$ intersect, and their intersection point lies on the ellipse $C$.

1. Find the equation of the ellipse $C$. 2. Let $E:\dfrac{x^2}{4a^2}+\dfrac{y^2}{4b^2}=1$. Let $P$ be any point on $C$; a line $y=kx+m$ through $P$ meets $E$ at points $A$ and $B$, and the ray $PO$ meets $E$ at a point $Q$. (a) Find $\dfrac{|OQ|}{|OP|}$. (b) Find the maximum area of $\triangle ABQ$.