The general form of the "Problem of Apollonius" is concerned with given any three circles, how to construct a circle tangent to all of the circles. So given circles a, b, and c, how to find a circle u such that u is tangent to a, b, and c.
If the radii of circles A and B are increased by the same amount such that the circles become tangent at a point K, then the center of u has not changed. So it is equivalent to find the following circle.
Now to find the desired circle, I will use some properties of inversion. If objects are tangent, then they are tangent after an inversion. So I choose a case that is easy to deal with after the inversion, then invert back.
In the above picture, choose the center of inversion to be the intersection of the blue and red circles. Since each of these circles passes through the center of inversion, then these circles map to parallel lines and the green circled maps to a circle.
So now we need to find a circle that is tangent to parallel lines and the green dashed circle.
The two pink circles satisfy that requirement. The center of the left pink circle corresponds to the center that is tangent to the "inside of the three circles, and the right pink circle's center corresponds to center that circumscribes the three circles.