What exactly is family of circles?

Question

How is it defined? I know that it's a product of two equations, but what does it represent exactly?

Answer 1

Maybe this picture can help? All of these circles have their centre on the perpendicular bisector between the two points (the axis $p_y$ here), and therefore the family of circles can be indexed by this line.

Answer 2

It is nothing but the name for a set of circles which share 2 common points.

For example $$x^2+(y-q)^2= 1+q^2$$ is a circle for each $q$ and each circle goes through points $A(1,0)$ and $B(-1,0)$. So this is a family of circles. If you made a picture you will get something like Arnaud draw.