Tangent lines to circle

Tangent lines from a point to a circle calculator

Print tangent lines from a point to circle calculator

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 basic definitions

Example 1

Example 2

Tangent lines between circle and a point summary

Print circle point tangent lines summary

Tangent points from a point
(x_p , y_p) 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
x_1,2	Tangent points x coordinates
y_1,2	Tangent points y coordinates
x_i	line connecting point to circle center x intercept
y_i	line connecting point to circle center y intercept

The distance between the point (x_p , y_p) 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 (x_p , y_p) with circle center

equation:
x intercept:	x_i = 0
y intercept:	y_i = 0

Circle form:

(x − a)² + (y − b)² = r²

Line connecting point (x_p , y_p) with circle center

equation:
x intercept:
y intercept:

Circle form:

x² + y² + Ax + By + C = 0

Line connecting point (x_p , y_p) 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 (x_p , y_p ) 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 implicit 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 implicit derivation is:

The slope of the tangent line can be expressed by the points (x , y) and P as the tangent 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

Example 2 - Tangent lines to a circle

Find the equations of the tangent lines to the circle given by the equation: x² + y² + 4x + 6y − 21 = 0 from the point P(−4 , 5).

Example 2 - general tangency of two lines and a circle

A simple way to find the points S and Q is by applying geometric methods.

The distance from the point to the circle center is:

First, we shell find the center and the radius of the circle, by the procedure for completing the square:

We can see that the center of the circle is at (−2 , −3) and the radius is equal to: √34

Since point Q is common to the line and the circle, its x and y coordinates are the same and we have to solve the set of equations:

From the line equation after eliminating y we get:

substitute y into the circle equation.

x^2-2cx+c^2-A^2/B^2 x^2-2A(C+Bk)/B x-(4A^2)/B^2 (C+Bk)^2-r^2=0

x^2 (1-A^2/B^2 )-x(2c-2A(C+Bk)/B)+c^2-(4A^2)/B^2 (C+Bk)^2-r^2=0

this is a quadratic equation for x: