Find the Points of Intersection of Two Ellipses

This is tutorial on finding the points of intersection of two ellipses given by their equations.

Example 1

Find the points of intersection of the two ellipses given by their equations as follows:
x2 / 16 + (y + 1)2 / 4 = 1
x2 / 2 + (y + 2)2 / 12 = 1
Solution to Example 1:

We first multiply all terms of the first equation by 16 and all the terms of the second equation by - 2 to obtain equivalent equations:
x
2 + 4 (y + 1)2 = 16
- x
2 - (1/6) (y + 2)2 = - 2
We now add the same sides of the two equations to obtain a quadratic equation
4 (y + 1)
2 - (1 / 6) (y + 2)2 = 14
Multiply all terms by 6, group like terms and rewrite the equation as
23 y
2 + 44y - 64 = 0
Solve the quadratic equation for y to obtain two solutions
y ≈ 0.97 and y ≈ -2.88
We now substitute the values of y already obtained into the equation x2 + 4 (y + 1)2 = 16 and solve it for x to obtain the x values
for y ≈ 0.97 ; x values are given by: x ≈ 0.730365 and x ≈ -0.730365
for y ≈ -2.88 ; x values are given by: x ≈ 1.36788 and x ≈ -1.36788
The 4 points of intersection of the two ellipses are
( 0.730365 , 0.97) ; ( -0.73 , 0.97) ; (1.37 , -2.88) ; (- 1.36788 , -2.88)
The graph of the two ellipses given above by their equations are shown below with their points of intersection.

Points of intersection of two ellipses

More Links and References on Ellipses

Points of Intersection of an Ellipse and a line
Find the Points of Intersection of a Circle and an Ellipse
Equation of Ellipse, Problems.
College Algebra Problems With Answers - sample 8: Equation of Ellipse
HTML5 Applet to Explore Equations of Ellipses
Ellipse Area and Perimeter Calculator