﻿ Intersection of a circle and a line
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
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
 Circle form: x2 + y2 + 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 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. (x1,2 − a)2 + (y1,2 − b)2 = r2 y1,2 = m x1,2 + d
 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
Example: Find the tangent line equation at point (1 , 2) to the circle: x2 + y2 + 2x + 3y − 13 = 0
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 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:
 Tangent line slope: Tangent line equation: