$ ABCD$ is a rectangle of area 2. $ P$ is a point on side $ CD$ and $ Q$ is the point where the incircle of $ \triangle PAB$ touches the side $ AB$. The product $ PA \cdot PB$ varies as $ ABCD$ and $ P$ vary. When $ PA \cdot PB$ attains its minimum value, a) Prove that $ AB \geq 2BC$, b) Find the value of $ AQ \cdot BQ$.