If $S_1=0$ is a circle and $S_2 = 0$ is another circle and they intersect at two points- A and B, then the family of circles passing through A,B is represented as
$$S_1 + k*S_2=0$$
The proof in my book says that the above equation always represents a circle. Whatever the value of k, it always passes through A,B since they satisfy that equation. Hence this represents all circles that pass through A,B.
I get that proof in one direction i.e. the equation represents circles passing through A,B. What I am having trouble understanding is how do we know ALL circles that pass through the points of intersection can be represented in this form?
What if there are equations of circles passing through A,B that cannot be represented this way? Is that part obvious from this sketch of the proof in the book?