A Constant Slope Ratio

A line through the left focus $F$ of the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{5} = 1$ meets the ellipse at $M$ and $N$, and $A(1, 0)$ is a fixed point on the major axis. Lines $MA$ and $NA$ meet the ellipse again at $P$ and $Q$. Prove that the ratio of the slopes of lines $PQ$ and $MN$ is constant.