In this case let the circles intersect at B and F respectively, as shown in figure below:
Attachment:
Inkedintersecting.gif
Now line
AC is a perpedicular bisector of BF at point D and AB=15 and BC=25. Hence to find the distance we need to calculate AD and DC.
Now triangle ABD is right triangle AD=\(\sqrt{AB^2-BD^2}=\sqrt{15^2-10^2}=11.18\)
Now triangle CBD is right triangle CD=\(\sqrt{BC^2-BD^2}=\sqrt{25^2-10^2}=22.91\).
Hence total distance AC = 11.18+22.91=34.09.
The portion marked in red are properties of a circle.
I can't tell where you got the 10 from, it shouldn't be as simple as just subtracting 25 from 15 that doesn't make sense.