# How can I find the equation of the shape formed by the intersection of two ellipses?

I am trying to find the equation of shape formed by the intersection of two ellipses. I have foci of each ellipse. The pictorial representation of the problem is shown in the figure below.

Clicke to see figure: Ellipse intersection

How can I generate the equation of shape shown pink colour?

I don't have that much base in mathematics, so if my question has any problem, please provide me feedback.

Manu

• You will have to intersect the two ellipses. This is hardly a trivial matter, but a search on MSE should bring up a good number of methods. – Parcly Taxel Jul 2 '19 at 15:10
• You should tell us more precisely what you intend to do because there are several ways to describe this shape. – user65203 Jul 2 '19 at 15:16
• By the way, it is not enough to know the foci. – user65203 Jul 2 '19 at 15:17
• @ParclyTaxel: with the given geometry, finding the two intersection points is very simple. – user65203 Jul 2 '19 at 15:18

$$\begin{cases} \dfrac{(x-x_1)^2}{a_1^2}+\dfrac{y^2}{b_1^2}\le 1,\\ \dfrac{(x-x_2)^2}{a_2^2}+\dfrac{y^2}{b_2^2}\le 1. \end{cases}$$