We define two ellipses by the polynomial equations:
Ax^{2} + By^{2} + Cx + Dy + E = 0 
(1) 
Fx^{2} + Gy^{2} + Hx + Iy + J = 0 
(2) 
From eq. (1) we get the value of y: 

Substitute the value of y into equation (2) yields the equation:
After multiplying all terms by 4B^{2} and open paranthesis we get the equation
In order to simplepy the equation we will define the following constants:
δ = 
2BDI − 2D^{2}G + 4BEG − 4B^{2}J 
φ = 
4BCG − 4B^{2}H 
θ = 
4ABG − 4B^{2}F 
μ = 
2BI − 2DG 
And we get the equation: 

Taking the squares of bothe sides we get:
D^{2}μ^{2} − 4ABμ^{2}x^{2} − 4BCμ^{2}x − 4BEμ^{2} = x^{4}θ^{2} + x^{2}φ^{2} + δ^{2} + 2θφx^{3} + 2θδx^{2} + 2φδx
After arranging the equation by powers we get a forthe power of x which can be solved mathematically:
x^{4}θ^{2} + x^{3}2θφ + x^{2}(φ^{2} + 2θδ + 4ABμ^{2}) + x(4BCμ^{2} + 2φδ) + (δ^{2} − D^{2}μ^{2} + 4BEμ^{2}) = 0
