Consider, say, the equation of the family of circles passing through two given points, which's equation is given by $S + kL =0$, where S is the equation of the circle having the two given points as the ends of a diameter and L is the equation of the line passing through the two points.
My question is that what is the logic behind writing the family of curves in this manner in that how do we know this particular equation covers all possible circles passing through the two points?
My initial thoughts were to try to prove this by assuming another curve $T$ such that $S + kL +λ T=0$ and then say that since at the two given points S and L are 0, $T$ also must be 0. However I can also apply this logic to $S +λT=0$ and incorectly conclude that the equation of the family of circles is $ S=0$.