A Chord and a Diameter

Consider the ellipse $G \colon \dfrac{x^2}{2} + y^2 = 1$. A line $l$ through the left focus $F$, not lying along the $x$-axis, meets $G$ at $A$ and $B$; let $M$ be the midpoint of chord $AB$, and let the line $OM$ meet $G$ at $C$ and $D$.

1. If $l$ has slope $1$, find the slope of $OM$. 2. Is there a line $l$ such that $|AM|^2 = |CM|\cdot|DM|$? If so, find its equation; if not, explain why.