A tutorial on how to find the points of intersection of two circles given by their equations; general solution.

Example 1

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.

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