Double Chord

Relevant wiki: Power of a Point

Let $EF$ be a diameter of the larger circle that passes through $B$ .

Note that by Power of a Point,

$AB \times BD = EB \times BF$ and $EB = 20 - 18; BF = 20 + 18$ .

So $AB \times BD = EB \times BF = (38)(2) = \boxed{76}$ .

$QC^{2}=18^{2}-OQ^{2}$

$QA^{2}=20^{2}-OQ^{2}$

$AB = QA - QC$

$BD = QC + QA$

$AB\cdot BD = (QA-QC)(QA+QC) = QA^{2} - QC^{2}$

$= (20^{2} - OQ^{2}) - (18^{2} - OQ^{2})$

$= 20^{2} - 18^{2} = (20+18)(20-18) = 38\cdot 2 = \boxed{76}$