The problem is this: what is the largest possible distance between a point on the circle (x - 3)^2 + y^2 = 4 and (x - 1)^2 + (y + 3)^2 = 25?

When I saw the problem, I immediately assumed it would be the distance between the circles plus the sum of the radii. This gives sqrt(13) + 7. However, the answer key said it was 12. Considering that sqrt(13) + 7 = 10.6055... which was relatively close to 12, I assumed there was a way of obtaining an even larger distance from two points that do not lie on the straight line connecting the radii. However, I could not figure out what these points were and to prove the distance was 12. Are there any ideas?

This question is from the PROBE I exam from 2011. Here is the solution guide, which does not show how they got their answers.

http://www.math.unl.edu/programs/mat...1probe1sol.pdf

Solution guides from other years do have the complete answer, such as the one from just last year, and I took this test in November.

http://www.math.unl.edu/programs/mat...ons_1111_4.pdf

These questions are for a mathematics competition for high school students in Nebraska (where I live).