Find the Points of Intersection of a circle and an ellipse

A 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 intersection 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


More References and links

Tutorials on equation of circle.
Tutorials on equation of circle (2).
Interactive tutorial on equation of circle.

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