A Tangent Intersection Locus

Let $P(x_0, y_0)$ be a point inside the ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). A line $l$ through $P$ meets the ellipse at $A$ and $B$, and the tangent lines to $E$ at $A$ and $B$ meet at $Q$. Find the equation of the locus of $Q$ as $l$ rotates about $P$, and find the minimum area of $\triangle QAB$.