# What will be the value of $r$?

Question: Two Circles of radius 36 and 9 touch each other externally, a third circle of radius r touches the two given circles externally and also their common tangent, then the value of r is ?

My attempt: So, I tried taking a Circle $A$ at orgin (radius = 36 units) and then a circle B at center(45,0). I tried the classical approach of finding the length of common tangent by dropping foot of perpendicular and stuff, but that doesn't really works out.

In my second attempt, I tried finding the equation of common tangent: $$y-y_o=m(x-x_o)+r \sqrt{1+m^2}$$ I successfully derived the value of $m^2$ but then faltered at the step as what should be the calculation to get the Center of my new little circle with radius $r$?

The answer given here is 4 units but I have no idea how. Please suggest your solutions.

Cheers!

## 1 Answer

Descartes' Circle Theorem (in the special case that the 4th circle is a line) provides an answer. You can calculate the curvature of each of the circles by $k_i = \frac{1}{r_i}$ and then use $k_{4}=k_{1}+k_{2}\pm 2{\sqrt {k_{1}k_{2}}}$ to calculate the curvature of the 3rd circle. Notice that $k_4$ is the curvature of the third circle because we actually have a fourth circle (whose curvature is associated with $k_3 = 0$) that is just the tangent line.

• This doesn't provide an answer. This is a link only. – Harsh Kumar Apr 15 '17 at 16:50
• While it is nice to be able to give a link for detailed history, etc., the relevant content of the link should be summarized (ideally, the most relevant information should be quoted with attribution). This allows Readers to better judge whether a link is worth following, and in the all-too-common cases where links go bad, it helps to have enough information to be able to reconstruct or replace the links. – hardmath Apr 15 '17 at 19:06