Intersection of circle and a line
Line
Line form:         y = mx + b y = x +
Line form:         Ax + By + C =0 x + y + = 0
Circle
Circle form:  (x-a)2 + (y-b)2 = r2 ( x - )2 + ( y - )2 = 2
Circle form:  x2 + y2 + Ax + By + C x2 + y2 + x + y + = 0
Intersection coordinates (x , y):
Intersection coordinates (x , y):
Distance between intersection points:
Distance of the line from circle center:
Line equations summary Circle equations summary
Intersection of circle and a line
Intersection points (x1 , y1) and
(x2 , y2) of a circle and a line of the form:      y = mx + d
Circle line intersection
Example: Find intersection points of circle:   (x − 3)2 + (y + 5)2 = 9
and the line:     y = −x + 1
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:
x1,2 = 6, 3    and    y1,2 = -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:   x2 + y2 + 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      x1,2 = 0.4, -2
Calculate   y   points:
∂ = 2.25     x1,2 = -2.8, -4
Circle form: Circle form
distance circles centers
distance circles centers
distance circles centers
Circle form: Circle form
x points

y points

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.
(x1 − a)2 + (y1 − b)2 = r2
y1 = mx1 + d
Check points on the circle
Line tangent to circle
Intersection of a line tangent to the circle at point (xt , yt) and a line of the form:   y = mx + d
Tangent circle
Example: Find the tangent line equation at point (1 , 2) to the circle: x2 + y2 + 2x + 3y − 13 = 0
Slop m
Line equation
And the tangent line equation is:
7y = -4x + 18
Circle form: Circle form
Tangent line slope: Tangent line slop
Tangent line equation: Tangent line equation
Provided that b ≠ yt
If   b = yt    then the line equation become:      x = xt
Circle form: Circle form
xt , yt   should be a point on the circle therefore:
Circle equation at tangent point
Tangent line slope: Tangent line slop
Tangent line equation: Tangent line equation