# 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. 