An intuitive explanation about family of circles. when we are finding a circle passing through point of intersection of two circles $S_1$ and $S_2$ we represent the family of family of circle as $S_1+kS_2=0$ where $k$ is a real constant.
when we need a circle passing through point of intersection of $S_1$ and line $L$ we represent it as $S_1+kL=0$.
my question is why does writing it like that work, what does $k$ exactly mean there. I understand when we are talking about a pair of intersecting lines we find family of lines $L_1$ and $L_2$ by $L_1+kL_2=0$. Writing family of lines makes sense perfectly to me because it represents linear combination of vectors in vector space or simply we are representing the span of the two vectors. But I don't really understand why does the same thing work for circles? I can't really imagine circles in a vector space or their linear combination. What I am searching for is an intuitive explanation for the two cases I mentioned, although the two cases can be proven by simply saying the locus of the combination satisfies with the intersecting circle, to which I agree and it can be proven rigorously too.
But I need a visual explanation of it, I want to really know why it works. It would be great if someone could explain it!


 A: The general equation of a circle in the plane in Cartesian
coordinates is
$$ f(x,y) := a(x^2+y^2) + bx + cy + d = 0 $$ where
$\,a,b,c,d\,$ are real constants. The surface
$\, z = f(x,y) \,$ is a paraboloid of revolution.
The intersection of this surface with the plane
$\,z=0\,$ is a circle in that plane. If
$\,g(x,y) := bx + cy + d,\,$
then the surface $\, z = g(x,y)\,$
is a plane and the
intersection of this with the plane $\,z=0\,$ is a line in that plane. Thus, if we denote
$$ S:=f(x,y),\quad L:= g(x,y), $$
then $\, S+L=0 \,$ is also the same equation of
the circle.
An equivalent way to think of this is that
$\ S=-L \,$ represents the intersection the
parboloid $\,z = S\,$ and the plane $\,z = -L.\,$
Thus, the same is true if we replace $\,L\,$ with
$\,kL\,$ where $\,k\ne 1.\,$ Now the equation
$\, kL = 0\,$  represents the same line, but the
plane $\, z = kL\,$ represents another plane
passing through the same line but with a different
slope. Thus $\, S + kL = 0\,$ is the equation of another
circle representing the intersection of the paraboloid
and a variable plane.
