Find the Points of Intersection of a Circle with a Line

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

Example 1

Find the points of intersection of the circle with the line given by their equations
(x - 2)
2 + (y + 3)2 = 4
2x + 2y = -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.

Points of intersection of a circle and a line

Figure 1. Intersection of a circle and a line.

More References and links

Step by Step Math Worksheets SolversNew !
Find Points Of Intersection of Circle and Line - Calculator.
Tutorials on equation of circle.
Tutorials on equation of circle (2).
Interactive tutorial on equation of circle.