An Intersection on a Fixed Line

Consider the ellipse $\dfrac{x^2}4+y^2=1$ and the hyperbola $\dfrac{x^2}4-y^2=1$. From the left vertex $P(-2,0)$, two lines are drawn meeting the ellipse at $A$, $C$ and the hyperbola at $B$, $D$. The lines $AD$ and $BC$ meet at a point $Q$. Prove that $Q$ lies on a fixed line.