Find the Points of Intersection of a Circle with a Line




This is 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 vy 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



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