I found in the internet a graph for the family of circles: $x^2+y^2=2cx $ and I'm not sure I quite understood if its true or I would be able to draw it by myself. this is the graph:
Aren't the center of the circles should be in $(0,0)$? and I'm aware of the symmetry because of the negative and positive values. but if this is the right graph, why is it really true? is there something that I'm missing?
The centers of the circles should be $(c, 0)$, which you can see if you rewrite your equation as follows: $$ (x-c)^2 + y^2 = x^2-2xc + c^2 + y^2 = (x^2-2xc + y^2) + c^2 = c^2 $$