A Constructed Tangent Line

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has focal distance $4$ and passes through the point $P\left(\sqrt 2,\sqrt 3\right)$.

1. Find the equation of the ellipse $C$. 2. Let $Q(x_0,y_0)$ (with $x_0y_0\neq 0$) be a point on $C$. Drop the perpendicular from $Q$ to the $x$-axis, with foot $E$. Take $A\left(0,2\sqrt 2\right)$, join $AE$, and draw the line through $A$ perpendicular to $AE$, meeting the $x$-axis at a point $D$. Let $G$ be the reflection of $D$ across the $y$-axis, and draw the line $QG$. Must the line $QG$ constructed this way always have exactly one common point with the ellipse $C$? Justify your answer.