Touchy Circles Part 2

Geometry Level 3

Circles $A, B,$ and $C$ are placed in the first quadrant so that circle $A$ is tangent to both the $x$ -axis and the $y$ -axis. Circle $B$ is tangent to the $x$ -axis and is externally tangent to circle $A$ , while circle $C$ is tangent to the $y$ -axis and is externally tangent to $A$ .

If a line passes through the centers of all three circles and the radii of circles $B$ and $C$ are equal, then how many times larger is the radius of circle $A$ than the radius of circle $B$ ?

4\sqrt{2}

3 + 2\sqrt{2}

4

6 - 2\sqrt{2}

1 solution

Chung Kevin
Sep 29, 2015

The slope of such a line must be $-1$ . Otherwise, the line will not pass through the centers of each circle. The right triangle whose hypotenuse is defined by the centers of circles $A$ and $B$ has sides of length $R - r$ , $R - r$ , and $R + r$ . This gives rise to the equation:

$(R - r)^2 + (R - r)^2 = (R + r)^2 \longrightarrow 2(R - r)^2 = (R + r)^2 \longrightarrow \sqrt{2}(R - r) = R + r$

With a little bit of algebra, we can achieve $R(\sqrt{2} - 1) = r(\sqrt{2} + 1) \longrightarrow \frac{R}{r} = \frac{\sqrt{2} + 1}{\sqrt{2} - 1} = 3 + 2\sqrt{2}$

Touchy Circles Part 2

1 solution

0 pending reports