Given two circles with the same origin, and:
- their radii are $r_1$ and $r_2$
- radii difference is $s=|r_2-r_1|$
- $(x_0,y_0)$ is an arbitrary point on the inner circle (that is, on the one with smaller radius, though it's also possible that $r_1=r_2$)
Extending two straight lines (i.e. one with no slope and one with slope of $0$) from $(x_0,y_0)$ find where these intersect the outer circle. Let's call the closest intersection point of the vertical line $(x_1,y_1)$ (note that $x_1=x_0$) and of the horizontal line $(x_2,y_2)$ (note that $y_2=y_0$). Other intersection points can be gotten by simply rotating these around the axes, so you can just ignore them in this problem.
This is the best visual representation I could draw: here.
This is based on a practical problem I have but I think I've included all the relevant information.