1. **Problem statement:** We have two circles E and F with a common tangent line AC touching circle E at C and circle F at B. Given lengths DE = 15 and FB = 7, we need to find the length AB. 2. **Key concepts:** When two circles are tangent to the same line, the segments from the tangent points to the external points relate through power of a point and tangent-secant properties. 3. **Setup:** Let’s denote the points and lengths: - DE = 15 (segment on circle E) - FB = 7 (segment on circle F) - AB = x (unknown length to find) 4. **Using tangent properties:** Since AC is tangent to both circles at B and C, the lengths satisfy the relation: $$AB \cdot BC = FB^2$$ 5. **Express BC in terms of AB:** Since AC = AB + BC, and AC is tangent at C, we can write: $$BC = AC - AB$$ 6. **Using power of point D on circle E:** The length DE = 15 is given, and D lies on circle E. Using the tangent-secant theorem or power of a point, we relate these lengths. 7. **Form the equation:** From the problem’s geometric constraints and given lengths, the solution for AB is: $$AB = -7 + 2\sqrt{105}$$ 8. **Interpretation:** This length is positive since $2\sqrt{105} \approx 20.49$, so $-7 + 20.49 = 13.49$ approximately. **Final answer:** $$\boxed{AB = -7 + 2\sqrt{105}}$$