An Affine Area Maximum

Consider the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$, points $A, B$ on it with $A$ in the first quadrant, and $O$ the origin; let $S$ be the area of $\triangle OAB$.

1. If $A = \left(1, \dfrac{3}{2}\right)$, find the maximum value of $S$. 2. Let $A(x_1, y_1)$, $B(x_2, y_2)$ with $3y_1 + y_2 = 0$. Find the equation of the line $AB$ when $S$ is maximized.