I got something that I think is trigonometrically equivalent (after more time and some math errors), deriving it as a law-of-cosines problem relating the distances from the center and the sum of the two radii. The quadratic has two roots which are reciprocals, because it doesn't distinguish as to which is the inner circle. An interesting problem; it almost looks like a Descartes' theorem problem, but the circle positioning makes that not apply.
I didn't think about the interpretation of the other root, just that it was obviously less than 1 so not correct for this problem. Your reading makes a lot of sense.When I got this solution, I poked at it with trig identities. You can write it in other forms, but I didn't find any which are obviously better.
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