# 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. More links and references related to the above topics.
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