A tutorial on finding the points of intersection of a circle with a line; general solution.

Example 1

Solution to Example 1

• We first solve the linear equation for y as follows:
  y = - x - 1/2
  
• We now substitute y in the equation of the circle by - x - 1/2 as follows
  (x - 2)² + (- x - 1/2 + 3)² = 4
  
• We now expand the above equation and group like terms
  2 x² - 9 x + 25/4 = 0
  
• Solve the above quadratic equation for x to obtain two solutions
  x = (9 + √(31)) / 4 and x = (9 - √(31)) / 4
  
• We now substitute the values of x already obtained into the equation y = - x - 1/2 to obtain the values for y as follows
  y = (-11 - √31) / 4
  and y = (-11 + √31) / 4
  
• The two points of intersection of the two cirlces are given by
  ((9 + √(31)) / 4 , (-11 - √31) / 4 ) and ((9 - √(31)) / 4 , (-11 + √31) / 4)
  Approximated as:(3.64 , - 4.14 ) and (0.86 , -1.36)

Shown below is the graph of the circle, the line and the two points of intersection.

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