A Constant Product of Slopes?

The ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has eccentricity $\dfrac{\sqrt{2}}{2}$, and the circle whose diameter is the minor axis of $C$ is tangent to the line $y = ax + 6$.

1. Find the standard equation of the ellipse $C$. 2. The line $l \colon y = k(x-1)$ (with $k \neq 0$) meets $C$ at two points $A$ and $B$. Through a point $P$ on $C$, a line parallel to the $x$-axis meets segment $AB$ at a point $Q$. Let $k'$ be the slope of line $OP$ (where $O$ is the origin), and let $S_1, S_2$ be the areas of triangles $APQ$ and $BPQ$. If

determine whether $k \cdot k'$ is a constant, and justify your answer.