# Thread: Calculate Tangent Line to 2 Circles

1. ## Calculate Tangent Line to 2 Circles

I am trying to calculate the equation of line tangent to 2 circles. I have attached a picture to give a better idea.

Circle 1 has a center at (3,4) and radius of 5
Circle 2 has a center at (9,19) and radius of 11,

So far I have wrote the equation = of the 2 circles, and found the distance between the points the tangent lines touch on each circle.

My picture is not the most acurate one, it is just to give an idea.

Thanks for the help

2. ## Geometry of tangents

First, remember that a tangent to a circle is perpendicular to a radial line segment to the point of tangency. Thus, recalling how we define the distance between a point and a line, the distance between a line tangent to a circle and the center of that circle is the radius of that circle. For a line of the form $\displaystyle Ax+By+C=0$, the distance to the point $\displaystyle (x_0,y_0)$ is $\displaystyle d=\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$
Since the centers of the circles are (3,4) and (9,19), and the radii are 5 and 11, respectively, you have
$\displaystyle \frac{|3A+4B+C|}{\sqrt{A^2+B^2}}=5$
and
$\displaystyle \frac{|9A+19B+C|}{\sqrt{A^2+B^2}}=11$

Note you will have an unneeded degree of freedom (as you could multiply the line equation $\displaystyle Ax+By+C=0$ by a constant and still have the same line), so you can pick A,B,C to be scaled such that $\displaystyle A^2+B^2=1$, so you then can define $\displaystyle \theta$ such that $\displaystyle A=\cos\theta,\,\,B=\sin\theta$, and our equations are
$\displaystyle |3\cos\theta+4\sin\theta+C|=5$
and
$\displaystyle |9\cos\theta+19\sin\theta+C|=11$

Thus you have two equations for two unknowns. Note that there will be four solutions: two will be the 'external' tangents, and the other two will be 'internal' tangents (which cross each other between the circles)

--Kevin C.

3. Originally Posted by CaliMan982
I am trying to calculate the equation of line tangent to 2 circles. I have attached a picture to give a better idea.

Circle 1 has a center at (3,4) and radius of 5
Circle 2 has a center at (9,19) and radius of 11,

So far I have wrote the equation = of the 2 circles, and found the distance between the points the tangent lines touch on each circle.

My picture is not the most acurate one, it is just to give an idea.

Thanks for the help
Hello,

I've attached a slightly more accurate sketch.

The steps to do the construction:

1. Draw a circle around the center of the green circle with radius $\displaystyle r_{green}-r_{blue}$
2. Draw a circle around the midpoint of $\displaystyle M_{green}M_{blue}$
3. Connect the intersection points of the circle of #2 with the circle of #1 with $\displaystyle M_{blue}$. The tangent you are looking for is a parallel of this line.
4. Translate this line by $\displaystyle r_{blue}$ units until it touches both circles.

Calculating the angles:

$\displaystyle \tan(\alpha) = \frac{15}6 = \frac52$

For symmetry reasons the angles $\displaystyle \beta_1$ and $\displaystyle \beta_2$ must be equal:

$\displaystyle \tan(\beta) = \frac6{15} = \frac25$

Calculating the slop of the tangents (red):

Use $\displaystyle \tan(\alpha + \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha) \cdot \tan(\beta)}$

Now you have to calculate the coordinates of the touching points on one of the circles to complete the equation of the tangent.

4. where does the 16/5 com from?

5. Originally Posted by CaliMan982
where does the 16/5 com from?
Pardon?

If you mean $\displaystyle \frac{15}{6}$ then it is the slope between the 2 centres of the circles:

$\displaystyle \frac{19-4}{9-3}=\frac{15}{6}$

6. How would I calculate a point on the cirlce the tangent line touches, i just calculated the slope.

7. Originally Posted by CaliMan982
How would I calculate a point on the cirlce the tangent line touches, i just calculated the slope.
Have a look here: http://www.mathhelpforum.com/math-he...348-post1.html

8. ## Complete solution to the four tangent lines of two circles

Tangents to Two Circles gives complete expressions for cos theta, sin theta, and "C" in terms of the coordinates of the centers and radii of the two circles, which you may find helpful.