Find the Points of Intersection of a circle and an ellipse




This is tutorial on finding the points of intersection of a circle and an ellipse given by their equations.

Example 1: Find the points of intersection of the circle and the ellipse given by their equations as follows:

x2 + y2 = 4
x2 / 4 + (y - 1)2 / 9 = 1

Solution to Example 1:

  • We first multiply all terms of the second equation by -4 to obtain:
    x2 + y2 = 4
    -x2 - (4 / 9) (y - 1)2 = - 4


  • We now add the same sides of the two equations to obtain a linear equation
    y2 - (4 / 9) (y - 1)2 = 0

  • Which may be written as
    5y2 + 8y - 4 = 0



  • Solve the quadratic equation for y to obtain two solutions

    y = -2 and 2/5

  • We now substitute the values of y = - 2 already obtained into the equation x2 + y2 = 4 and solve it for x as follows

    x2 + (-2)2 = 4

    x = 0

  • We now substitute the values of y = 2/5 already obtained into the equation x2 + y2 = 4 and solve it for x as follows

    x2 + (2/5)2 = 4

    x = 4 √6 / 5 ≈ 1.96 and x = - 4 √6 / 5 ≈ -1.96

  • The points of intesection of the ellipse and the circle are

    (-2 , 0) ; (-4 √6 / 5 , 2/5) ; (4 √6 / 5 , 2/5)

Shown below is the graph of a circle and an ellipse and their points of intersection.

Points of intersection of a circle and an ellipse



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