The following image shows an ellipse (orange) and two concentric circles (green) with centre at the centre of the ellipse:
The outer circle is the one with the major axis as its diameter and is called the Auxiliary circle. The inner circle's diameter is the minor axis of the ellipse. Do we have any special name for the inner circle as we have for the outer one?
I am unable to find any relevant information on the internet.
The term "auxiliary circle" does not seem to be in wide usage these days, but you can find it in various texts. Generally, the circumscribed and inscribed circles are referred to respectively as "the major auxiliary circle" and "the minor auxiliary circle" and generically as "auxiliary circles," according to Runkle's Plane Analytic Geometry (1888)
Some other views of the same book and topic on archive.org are here and here.
You can also take a look at Abbott's Practical Geometry, and search for "Practical Geometry" on archive.org to find books that use the term.