A Ratio-Preserving Chord

Let $A(2, 3)$ and $B(2, 1)$. The hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) has an asymptote $y = x$, and the chord cut by line $AB$ on $C$ has length $2\sqrt{3}$.

1. Find the equation of $C$. 2. Let $P, Q$ be distinct points on the right branch of $C$ such that $\overrightarrow{AP} = \lambda\overrightarrow{AQ}$ for some real $\lambda$.

1. If a point $D$ satisfies $\overrightarrow{PD} = \lambda\overrightarrow{DQ}$, prove that $D$ always lies on a fixed line.
2. If the line $PB$ meets $C$ again at $R$ (with $R \neq Q$), prove that the line $QR$ passes through a fixed point, and find that point.