Joel's Problem 4: Maximum of maximums

Consider a rectangle R with vertices A, B, C, D in clockwise order. Among all points P on segment CD, let the maximum value of $PA \cdot PB$ be $f (R)$ . Among all rectangles Q of area $\sqrt {2}+1$ , let $M$ be the maximum value of $f (Q)$ . However, only some rectangles can have a point on segment CD such that $PA \cdot PB=M$ . Let $S$ be the set of all such rectangles. Find the maximum possible value of $\frac {AB}{BC}$ over all rectangles in $S$ .