﻿ Tangent lines from a point to circle
 Tangent lines from a point to circle ▲
 Circles form:   (x-a)2 + (y-b)2 = r2 ( x - )2 + ( y - )2 = 2
 Circles form:   x2 + y2 + Ax + By + C = 0 x2 + y2 + x + y + = 0
 Point outside the circle:     input (x, y): ( , )
 Circle equation: Tangent line 1 Equation: Tangency point: Tangent line 2 Equation: Tangency point: Distance from point to circle center: Line connecting circle center to the point Line equation: Line angle: Degree x axis intercept: y axis intercept:
 Circle equations summary Line Geometry Example 1
 Tangent lines between circle and point ▲
Tangent points from a point
(xp , yp) on a circle.
Example: Find the tangent points on the circle (x − 2)2 + (y + 5)2 = 9   from point (7 , 1).
In this case:    a = 2    b = − 5    r = 3
The value of the square root is:
x1 = 4.87         x2 = 0.61
y1 = − 5.89         y2 = − 2.34
So first tangency point is:
(4.87,-5.89) and the second point is the other points: (0.61,-2.34)
Now we can check if the tangent point that we found is on the circle:
(4.866-2)2 + (-5.888 + 5)2 =
2.8662 + (-.888)2 = 9
Note: we used higher precision of the point coordinate otherwise we would get slightly different value then 9.

nomenclature:
 D − Distance from point to circle center d − Distance from point to tangent point θ − Angle between the two tangent lines x1,2 Tangent points x coordinates y1,2 Tangent points y coordinates xi line connecting point to circle center x intercept yi line connecting point to circle center y intercept
The distance between the point (xp , yp) and the tangent point (1) is:
The angle between the two tangent lines   θ   is:
Note: in the equations above x1 can be replaced by x2.
 Circle form: x2 + y2 = r2
Line connecting point (xp , yp) with circle center
 equation: x intercept: xi = 0 y intercept: yi = 0
 Circle form: (x − a)2 + (y − b)2 = r2
Line connecting point (xp , yp) with circle center
 equation: x intercept: y intercept:
 Circle form: x2 + y2 + Ax + By + C = 0
Line connecting point (xp , yp) with circle center
 equation: x intercept: y intercept:
If two points (x1 , y1)   and (x2 , y2) on the circle are given then the intersection point (xp , yp ) created by the tangents lines is:
 Where: and
 Example 1 - Circle tangent lines ▲
Find the equations of the line tangent to the circle given by:  x2 + y2 + 2x − 4y = 0  at the point  P(1 , 3).
The incline of a line tangent to the circle can be found by inplicite derivation of the equation of the circle related to x (derivation dx / dy)
If the equation of the circle is:
Ax2 + Ay2 + Bx + Cy + D = 0
Then the explicite derivation is:
The slope of the tangent line can be expressed by the points (x , y) and P as the tangente of  θ  or  m  as:
 (tangent of the line) y − 3 = mx − m mx − y + 3 − = 0
And the equation of the tangent line is:      2x + y − 5 = 0