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 Print content ▲
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
Line form: y = m x + d
Circle form: (x − a)2 + (y − b)2 = r 2
distance circles centers
distance circles centers
distance circles centers
Circle form: x2 + y2 + A x + B y + C = 0
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 does not intersect 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.
Check points on the circle
(x1,2 − a)2 + (y1,2 − b)2 = r2 y1,2 = m x1,2 + d
Line tangent to circle Print content ▲
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
Slope m
Line equation
And the tangent line equation is:
7y = -4x + 18
Line form: y = m x + d
Circle form: (x − a)2 + (y − b)2 = r 2
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: x2 + y2 + A x + B y + C = 0
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