$ 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$.