E is the external similarity point of the two circles.
K is the internal similiarity point.

Notes:

Given d, r and s the ratio a/(d+a) = r/s gives a unique solution,
but it is an incomplete question.
a appears in the numerator and denominator in such a way that, if we change the sign of a, the sign of a/(d+a) does not change.
To find a solution on the same side of B, but on the other side of A, we must ask explicitily:
a'(d-a') =? r/s
There is such an a'.

solving for a' in terms of a and d, we get
a' = ad/(2a + d)
where a', a and d are all positive values (lengths). This gives the range restrictions:
a>a'
d> 2a'
Assume A is the smaller circle.
Then these conditions are the same thing as saying
-both intersections lie on the same side of B as A
- E is on the far side of A,
-K falls between A and B, and is closer to A.
These may or may not already be obvious, depending on how you frame the problem.