A Fixed Point from Tangents to a Shrinking Circle

As shown in the figure, let $A$ be the top vertex of the ellipse $\dfrac{x^2}{4} + y^2 = 1$. The two lines through $A$ tangent to the circle $M: (x+1)^2 + y^2 = r^2$ (where $0 < r < 1$) meet the ellipse again at points $B$ and $D$, both different from $A$. As $r$ varies, does the line $BD$ always pass through a fixed point? If so, find this fixed point; if not, explain why.