2014 HMMT November General Round Problem 1

Two circles \omega and \gamma have radii 3 and 4 respectively, and their centers are 10 units apart. Let x be the shortest possible distance between a point on \omega and a point on \gamma, and let y be the longest possible distance between a point on \omega and a point on \gamma. Find the product xy.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2014 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.