I've been told that most of the time, when using geometric inversion (which is basically just reflecting across a circle), the radius of the circle used for inversion doesn't matter, because changing the radius of the circle only dilates the result.
When does the length of the radius actually matter?
Depends on why you are using inversion. One common example is that you want to exchange the inside and outside of a circle while leaving the circle fixed. This is exactly similar to reflection across a line. The reflection exchanges the two sides of the line while leaving the line fixed. Any reflection will have a fixed set and often you want a particular set to be fixed.
Another example is Apollonian circles where the two circle centers are required to be exchanged by a circle inversion. In this case there is a whole family of circles to accomplish this, but if the center of the inverting circle is fixed, then a unique circle inversion is required to exchange them.