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: 
(x − a)^{2} + (y − b)^{2} = r ^{2} 
Circle form: 
x^{2} + y^{2} + A x + B y + C = 0 
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: Because during the solution we get two values for x coordinate and two values for y coordinate,
it is important to match the correct values for x and y. there can be 4 possible sets of points (red points)
but there are only two correct intersection points (green points) in order to match the correct points we can
substitute the points x and y into the circle and line equation and see which of them are satisfying the
equations.


(x_{1,2} − a)^{2} + (y_{1,2} − b)^{2} = r^{2} 
y_{1,2} = m x_{1,2} + d 
