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:
    x2 + 4 (y + 1)2 = 16
    - x2 - (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 y2 + 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 obatain 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)

Shown below is the graph of two ellipses and their points of intersection.

Points of intersection of two ellipses



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