Help with geometry problem

[Guest]

Hello, I am looking for some help with a geometry problem I have come across. The problem involves two circles with radius r1 and r2 respectively. A line is tangent to both of them. I am trying to find the angle (alpha) as described in the picture. I would know r1, r2, and the distance between the centers of the circles (c). The project this problem has arisen in will eventually culminate in r2 decreasing as a function of time and trying to find alpha at time t.

[shyamjayakannan]

Let the centers be [tex]C_1[/tex] and [tex]C_2[/tex]. Denote the point of intersection of the tangent and the line joining the centers be [tex]D[/tex].

Now, [tex]c=C_1D+C_2D=\frac{r_1}{\cos\alpha}+\frac{r_2}{\cos\alpha} \Rightarrow\boxed{\alpha=\cos^{-1}\left(\frac{r_1+r_2}{c}\right)}[/tex]