CWMO 2001 Problem 2

$ABCD$ 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 $AQ \cdot BQ$ .