Equation general solution of intersection of two elipse

Question

I have two elipse.

E1: $\dfrac{(x-x_1)^2}{a^2}+\dfrac{(y-y_1)^2}{b^2}=1$ and

E2: $\dfrac{(x-x_2)^2}{c^2}+\dfrac{(y-y_2)^2}{d^2}=1$.

Please help me what is Equation general solution of intersection of this two elipse.

$x_1=?$

$x_2=?$

$x_3=?$

$x_4=?$

Thanks so much.

Answer 1

Consider a nice situation where there are four real solutions:

$ M2
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : R=QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I=ideal(x^2+((y-2)/4)^2-1,((x-1)/9)^2+(y-2)^2-1)

             2    1 2   1    3   1 2    2    2         244
o2 = ideal (x  + --y  - -y - -, --x  + y  - --x - 4y + ---)
                 16     4    4  81          81          81

o2 : Ideal of R

i3 : ideal gens gb I

                 2                            2
o3 = ideal (1295y  - 32x - 5180y + 3916, 1295x  + 2x - 1216)

o3 : Ideal of R

In the gröbner basis there is $1295x^2+2x-1216=0$ which gives you two $x$-values to put into the other generator $1295y^2 - 32x - 5180y + 3916=0$. Now you have your four points.

As you can see the explicit solution is hard to write down even for a nice example. When the ellipses don't meet, you often need to consider projective complex geometry to get all four solutions; which means homogenising the equations with a third variable $z$. Consider concentric circles:

$ M2
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : R=QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : I=ideal(x^2+y^2-z^2,x^2+y^2-2*z^2)

             2    2    2   2    2     2
o2 = ideal (x  + y  - z , x  + y  - 2z )

o2 : Ideal of R

i3 : ideal gens gb I

             2   2    2
o3 = ideal (z , x  + y )

where you get the circular points at infinity with multiplicity.