Generalization of intersection of circles? Let us consider two circles in the (real) plane:
$C_1 : (x-x_1)^2 + (y-y_1)^2 - r_1^2 = 0$
$C_2 : (x-x_2)^2 + (y-y_2)^2 - r_2^2 = 0$
In order to calculate their intersection point we can easily find the line defined by the two points by subtracting $C_1$ from $C_2$:
$L: -2(x_1-x_2)x + (x_1^2-x_2^2) - 2(y_1-y_2)y + (y_1^2-y_2^2) - (r_1^2-r_2^2) = 0$
Plugging this back into either circle equation $C_i$ we can easily determine the intersection points of $C_1$ and $C_2$. So we can view the line $L$ as the line defined by the two intersection points.
But is there also a nice geometric interpretation of $L$ if the circles do not intersect?
(That is, when the distance between their centers is smaller than $\min \{r_1,r_2\}$ or larger than $r_1+r_2$.)

 A: if $C_1$ doesn't intersect $C_2$, then there is a pair of points $(A,B)$ such that $A$ is the image of $B$ by both inversions through $C_1$ and $C_2$.
Then the line you are seeing is a line going through the middle of those two points. Also, because the line joining the two centers is an axis of symmetry of the picture and since your line is defined purely geometrically, it has to stay an axis of symmetry, so your line has to be perpendicular to the axis (it can't be the axis itself or else it would have real intersection points with the circles).
There is a way to interpret $A$ and $B$ as the pair of conjugate complex points that are at the intersection of $C_1$ and $C_2$, (and so also on the line), I give a lot more info about that in this answer What is the reflection across a parabola?
That line is also the locus of points that have the same power with respect to both circles, see Line intersecting complex points of two circles
A: Pass from $\mathbb{R}^2$ to $\mathbb{C}^2$, and the equations will always have solutions.  In the case where the circles don't intersect, the points predicted by the equations will be imaginary.
Yet, the line between those imaginary points does intersect the real plane.  That is the line you see.
