I know that to obtain the intersection points between a line with one circle, we can use
$$x^2+y^2=R^2$$ and $$mx+c=y,$$ such that $$x^2(1+m^2)+2mcx+c^2-R^2=0,$$ where the solution of this quadratic equation gives the intersection points, but if I consider two concentric circles with a line intersecting each one of them, how this problem would be defined?
In short, I want to know how to find the points H, I, J, and G in the figure below 
Any tip, or reference I would thank you very lot.
Best Answer
What you derived is a quadratic equation, the roots of which are the $x-$coordinates of the intersections on the circle having radius $R$. Let the other circle have radius $r$. That leads to this equation in the same form:
$x^2(1+m^2)+2mcx+c^2-r^2 = 0$
If what you want is a single equation with the $x-$coordinates of all intersections, simply use the product, which results in this quartic equation:
$[x^2(1+m^2)+2mcx+c^2-R^2][x^2(1+m^2)+2mcx+c^2-r^2] = 0$
Expand it if you wish. It does not look particularly useful, but there it is.