The equation of an ellipse with focal points $(a_1,b_1)$ and $(a_2,b_2)$ which passes through the point $(a_3,b_3)$ is given by the equation $ \sqrt{ (x-a_1)^2+(y-b_1)^2 } + \sqrt{ (x-a_2)^2+(y-b_2)^2 } = \sqrt{ (a_3-a_1)^2+(b_3-b_1)^2 } + \sqrt{ (a_3-a_2)^2+(b_3-b_2)^2 }.$ Suppose that an ellipse $E_1$ is given by the equation $ \sqrt{ (x-a_1)^2+(y-b_1)^2 } + \sqrt{ (x-a_2)^2+(y-b_2)^2 } = \sqrt{ (a_3-a_1)^2+(b_3-b_1)^2 } + \sqrt{ (a_3-a_2)^2+(b_3-b_2)^2 } $ and another ellipse $E_2$ is given by the equation $ \sqrt{ (x-c_1)^2+(y-d_1)^2 } + \sqrt{ (x-c_2)^2+(y-d_2)^2 } = \sqrt{ (c_3-c_1)^2+(d_3-d_1)^2 } + \sqrt{ (c_3-c_2)^2+(d_3-d_2)^2 } $ and further assume that they intersect at two points. How does one figure out the coordinates of these two points? I've tried matlab for this problem but it just has a seizure. Even ideally I don't know the theory for how to do this. Further, is there a simple way to determine the equation of the line that passes between these two points?