Intersection points (x_{1} , y_{1}) and (x_{2} , y_{2})
of a circle and a line of the form: y = mx + d
Example: Find intersection points of circle: (x − 3)^{2} + (y + 5)^{2} = 9
Solution: In our case
m = − 1 d = 1 a = 3 b = − 5 r = 3
Calculate ∂ = 9 so the line intersects the circle at the points:
x_{1,2} = 6, 3 and y_{1,2} = 5, 2
And the intersection points are (6, 5) and (3, 2)
Correctness check:
For circle: 
(6  3)^{2} + (5 + 5)^{2} = 9 
For line: 
5 = 6 + 1 = 5 
Example: Find intersection points of circle: x^{2} + y^{2} + 3x + 4y + 2 = 0
and the line: x − 2y − 6 = 0
Solution: In this case
m = 0.5 d = 3 A = 3 B = 4 C = 2
Calculate x points:
∂ = 9 x_{1,2} = 0.4, 2
Calculate y points:
∂ = 2.25 x_{1,2} = 2.8, 4

Circle form: 

Circle form: 

If ∂ > 0 
then two intersection points exists 
If ∂ = 0 
then the line is tangent to the circle 
If ∂ < 0 
then the line do not intersects the circle 
Note: it is important to keep the order of the square sign in x as ± and y as ± otherwise we will get wrong
points (the red points in the drawing). Anyway if we are not sure of the correct pairs of the points we can check
them by substituting one of the intersection point into the circle and line equation.
(x_{1} − a)^{2} + (y_{1} − b)^{2} = r^{2}
y_{1} = mx_{1} + d


