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 $$\begin{gathered} \sqrt{ (x-a_1)^2+(y-b_1)^2 } + \sqrt{ (x-a_2)^2+(y-b_2)^2 }\\ = \sqrt{ (a_3-a_1)^2+(b_3-b_1)^2 } + \sqrt{ (a_3-a_2)^2+(b_3-b_2)^2 }.\end{gathered}$$ Suppose that an ellipse $E_1$ is given by the equation $$\begin{gathered} \sqrt{ (x-a_1)^2+(y-b_1)^2 } + \sqrt{ (x-a_2)^2+(y-b_2)^2 }\\ = \sqrt{ (a_3-a_1)^2+(b_3-b_1)^2 } + \sqrt{ (a_3-a_2)^2+(b_3-b_2)^2 }\end{gathered} $$ and another ellipse $E_2$ is given by the equation $$\begin{gathered} \sqrt{ (x-c_1)^2+(y-d_1)^2 } + \sqrt{ (x-c_2)^2+(y-d_2)^2 }\\ = \sqrt{ (c_3-c_1)^2+(d_3-d_1)^2 } + \sqrt{ (c_3-c_2)^2+(d_3-d_2)^2 }\end{gathered} $$ 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).
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$\begingroup$ Please don't use the tag elliptic-curves for ellipses. These are distinct topics (by the way, the set of real points of an elliptic curve in the plane is never an ellipse). $\endgroup$– YCorCommented Jan 31 at 13:52
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1$\begingroup$ Do you have a response to the answer below? $\endgroup$– Iosif PinelisCommented Feb 1 at 3:24
2 Answers
See this on the slightly more general matter of the intersection of two conic sections, with a further reference to Richter-Gebert's book.
This involves solving a cubic equation whose coefficients are polynomials (of degrees up to $4$ in at least $4$ variables), so that the resulting expressions are not pretty.
This answer is for the "radical axis" part of your question. In the theory of conics, the concept of the radical axis of two circles generalizes to a pair of common chords. A common chord of two conics is a line between two intersection points. Intersection points can be real or imaginary, and so the common chords can be real or imaginary, but there are always at least two real common chords, no matter how many of the intersection points are real.
In the case of two circles, the radical axis is a real line, whether or not the circles intersect in real points. And it is also a common chord, although in the case of non-intersecting circles it is a real line that connects two imaginary points of intersection. And it is part of a pair of lines, the other being the line at infinity, which also satisfies the definition of a radical axis.
A method for calculating common chords is given in Smith, Conic Sections, Article 234, pg. 253, shown below.
In the case of your question - two conics intersecting in two points - the method will yield a degenerate conic consisting of two lines, one of which is the line connecting the two points of intersection. In the end, it's not too different from subtracting the equation of one circle from that of another.
Update: So far we've seen how to calculate, without first calculating points of intersection, conics consisting of pairs of common chords. Article 37 from the same book (shown below) shows how to extract the individual lines.
Let
$$ ax^2+2hxy+by^2+2gx+2fy+c=0 $$
be a degenerate conic consisting of two straight lines. Then the lines can be written as
$$ ax+hy+g=\pm((h^2-ab)^{1/2}y+(g^2-ac)^{1/2}). $$
Article 37 also derives the zero-discriminant condition used in Article 234.
