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 }.$$$\begin{gathered} \sqrt{ (x-a_1)^2+(y-b_1)^2 } + \sqrt{ (x-a_2)^2+(y-b_2)^2 }\\ = \sqrt{ (a_3-a_1)^2+(b_3-b_1)^2 } + \sqrt{ (a_3-a_2)^2+(b_3-b_2)^2 }.\end{gathered}$$ 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 } $$$\begin{gathered} \sqrt{ (x-a_1)^2+(y-b_1)^2 } + \sqrt{ (x-a_2)^2+(y-b_2)^2 }\\ = \sqrt{ (a_3-a_1)^2+(b_3-b_1)^2 } + \sqrt{ (a_3-a_2)^2+(b_3-b_2)^2 }\end{gathered} $$ 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 } $$$\begin{gathered} \sqrt{ (x-c_1)^2+(y-d_1)^2 } + \sqrt{ (x-c_2)^2+(y-d_2)^2 }\\ = \sqrt{ (c_3-c_1)^2+(d_3-d_1)^2 } + \sqrt{ (c_3-c_2)^2+(d_3-d_2)^2 }\end{gathered} $$ 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? (For example, to find the radical line of two circles, you simply subtract one equation from the other, but this doesn't work in the case of ellipses).
