@divinedolphin

The cool trick here is to visualize a large right triangle, \(\bigtriangleup FDE,\) which you can get by sliding the tangent line \(\overline{AB}\) over until it touches the center of the smaller circle:

First, it's helpful to remember that the question states that the circles are \(9\) units apart at their closest point. This is equivalent to the yellow segment highlighted down below.

d1097186-1deb-423a-a3af-e675af4c8d0a-image.png

Conveniently, the line connecting the two circles' centers, \(\overline{DE},\) is just this segment of length \(9\) plus the radius of \(6\) and the radius of \(2,\) so this line \(DE = 6 + 9 + 2 = 17.\)

Now we don't even need to know the Pythagorean Theorem, since we know of a right triangle with a leg of \(8\) and a hypotenuse of \(17:\) it's the \(8 - 15 - 17\) right triangle! This means the other leg equal \(15,\) which happens to be the length that we want!