Gave the SAT exam recently and almost aced the Maths section. Almost because there was this one question I couldn't wrap my head around to solve. I don't remember the exact question, but it went something like this:
If circles of radius 6 cm are drawn with their centre being a tangential point on the surface of a sphere of radius 10 cm, is it possible that any points of these circles will intersect?
I tried to visualize it and the choice which seemed reasonable to me was 'No' which, now that I've received my results, I think was the wrong answer. The other choices were 'intersect at 1 point' and 'intersect at 2 points'. So please, could anyone mathematically explain what the answer to this question is?
Thanks!
Here's the kicker: all such circles lie on a sphere. Depending on the interpretation of the problem, it is either the sphere of radius $10,$ or a sphere of radius $2\sqrt{34}$ (by Pythagorean Theorem) having the same center as the sphere of radius $10.$ In either case, these circles are not "great circles" (that is, their radius is less than the radius of the sphere), so they don't have to intersect (any two distinct great circles must intersect at exactly two points), but they may intersect. If two such circles intersect at all, they will intersect at either one point or two points (if two circles intersect at three or more points, then they are the same circle), and both are possible.