# 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. More links and references related to the above topics.