A Harmonic Slope Invariant

The ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has eccentricity $\dfrac{1}{2}$. Two lines through the interior point $P(1, 1)$ meet the ellipse at $A, C$ and $B, D$ respectively, with

for a real $\lambda$. When $C$ is the right vertex of the ellipse, $\lambda = \dfrac{5}{7}$.

1. Find the equation of the ellipse. 2. As $\lambda$ varies, is the slope $k_{AB}$ of line $AB$ constant? If so, find it; if not, explain why.