Subtracting equations of two circles which don't intersect

Question

Suppose you subtract the equations of two circles that do not intersect. In doing so, you will obtain a line which is an "extraneous solution set".

Is there any geometric significance to this line?

Answer 1

The line is the radical axis or "power line" of the two circles.

Consider a circle with center $C=(h, k)$ and radius $r$. The power $p$ of point $P=(x,y)$ with respect to $\bigcirc C$, is given by $$p^2 = |\overline{PC}|^2 - r^2 \quad\to\quad p^2 = ( x - h )^2 + ( y - k )^2 - r^2$$

The radical axis of two circles is the set of points for which the power with respect to each circle matches. That is, given circles with centers $C_i = (h_i, k_i)$ and radii $r_i$, point $P=(x,y)$ is on the radical axis of $\bigcirc C_1$ and $\bigcirc C_2$ when $$( x - h_1 )^2 + ( y - k_1 )^2 - r_1^2 = ( x - h_2 )^2 + ( y - k_2 )^2 - r_2^2$$ Simplifying this equation (which cancels the $x^2$ and $y^2$ terms, leaving a linear result) is effectively the same process as subtracting the standard equation for $C_2$ from that of $C_1$ (or vice versa).