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 x^{2} - 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.