Find the Points of Intersection of two Circles

A tutorial on how to find 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 obtain 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 obtain
    (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 circles 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 obtained above.

Points of intersection of two circles

More References and links

Step by Step Maths Worksheets Solvers
Points of Intersection of Two Circles - Calculator.
Tutorials on equation of circle.
Tutorials on equation of circle (2).
Interactive tutorial on equation of circle.

Computer Technology Simply Explained

Search

Navigation

Keep In Touch