Find the Points of Intersection of two Circles




This is tutorial on finding the points of intersection of two circles given by their equations; general solution.

Example 1: Find the points of intersection of the circles given by their equations as follows:

(x - 2)2 + (y - 3)2 = 9
(x - 1)2 + (y + 1)2 = 16

Solution to Example 1:

  • We first expand the two equations as follows:
    x2 - 4x + 4 + y2 - 6y + 9 = 9
    x2 - 2x + 1 + y2 + 2y + 1 = 16

  • Multiply all terms in the first equation by -1 to abtain an equivalent equation and keep the second equation unchanged
    -x2 + 4x - 4 - y2 + 6y - 9 = -9
    x2 - 2x + 1 + y2 + 2y + 1 = 16

  • We now add the same sides of the two equations to obtain a linear equation
    2x - 3 + 8y - 8 = 7

  • Which may written as
    x + 4y = 9 or x = 9 - 4y

  • We now substitute x by 9 - 4y in the first equation to obatin

    (9 - 4y)2 - 4(9 - 4y) + 4 + y2 - 6y + 9 = 9

  • Which may be written as

    17y2 -62y + 49 = 0

  • Solve the quadratic equation for y to obtain two solutions

    y = (31 + 8√2) / 17 ≈ 2.49

    and y = (31 - 8√2) / 17 ≈ 1.16

  • We now substitute the values of y already obtained into the equation x = 9 - 4y to obtain the values for x as follows

    x = (29 + 32√2) / 17 ≈ - 0.96

    and x = (29 - 32√2) / 17 ≈ 4.37

  • The two points of intersection of the two cirlces are given by

    (- 0.96 , 2.49) and (4.37 , 1.16)

Shown below is the graph of the two circles and the linear equation x + 4y = 9.

Points of intersection of two circles



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Updated: 2 April 2013

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