This answer is for the "radical axis" part of your question.  In the theory of conics, the concept of the radical axis of two circles generalizes to a pair of common chords.  A common chord of two conics is a line between two intersection points.  Intersection points can be real or imaginary, and so the common chords can be real or imaginary, but there are always at least two real common chords, no matter how many of the intersection points are real.

In the case of two circles, the radical axis is a real line, whether or not the circles intersect in real points.  And it is also a common chord, although in the case of non-intersecting circles it is a real line that connects two imaginary points of intersection.  And it is part of a pair of lines, the other being the line at infinity, which also satisfies the definition of a radical axis.

A method for calculating common chords is given in [Smith, *Conic Sections*, Article 234, pg. 253][1], excerpted below.

[![enter image description here][2]][2] 

In the case of your question - two conics that intersect in two points - the method will yield a degenerate conic consisting of two lines, one of which is the line connecting the two points of intersection.  In the end, it's not too different from subtracting the equation of one circle from that of another.


  [1]: https://archive.org/details/cu31924063287340/page/n271/mode/2up
  [2]: https://i.sstatic.net/kKQto.jpg