Can there be two parameters in equation of family of circles passing through the points of intersection of a line and a circle?

Question

I always find it written as: S + kL = 0, (where k is the parameter).

But while playing in Desmos I found out the it can also be written as L + kS =0 or more generally , mS + kL = 0. (where both m and k are parameters). Am I right? If I am then why don't I find these forms anywhere?