Two Chords and a Fixed Slope Ratio

Consider the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$. A line $AB$ passes through the fixed point $Q(-1,0)$; let $P(1,0)$. The lines $PA$ and $PB$ meet the ellipse again at points $C$ and $D$ respectively. Prove that $CD$ passes through a fixed point, and that the ratio of the slopes of $CD$ and $AB$ is constant.