Conjugate Point Pairs

An ellipse $E$ centered at the origin $O$ with foci on the $x$-axis passes through $A\left(\sqrt{2}, -\dfrac{\sqrt{2}}{2}\right)$ and $B(0, 1)$.

1. Find the standard equation of $E$. 2. For an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$), call points $M(x_1, y_1)$ and $N(x_2, y_2)$ on it a conjugate pair $[M, N]$ if

1. Prove there are exactly two points $G$ such that $[A, G]$ is a conjugate pair of $E$, and find their coordinates.
2. Call these two points $G_1, G_2$. A line $l$ through $P(2, 1)$ meets $E$ at points $C$ and $D$. Is there a fixed point $Q$ on the line $G_1G_2$ such that the product of the slopes of $QC$ and $QD$ is constant? If so, find $Q$; if not, explain why.