Touchy Circles

Geometry Level 3

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

If the radius of circle $B$ is half the radius of circle $A$ , then what is the slope of the line passing through the centers of $A$ and $B$ ?

-2

-\frac{1}{2}

-2\sqrt{2}

-\frac{\sqrt{2}}{4}

1 solution

Chung Kevin
Sep 29, 2015

The right triangle formed above has hypotenuse of length 3R and one leg of length R. By the Pythagorean Theorem , the other leg has a length of R $\sqrt{8}$ . Using the formula slope $m = \frac{\Delta y}{\Delta x}$ , we see that as we move from the center of A to the center of B, the value of $y$ decreases by R, while the value of $x$ increases by R $\sqrt{8}$ , so the slope is $\frac{-R}{R\sqrt{8}} = -\frac{1}{\sqrt{8}} = \boxed{-\frac{\sqrt{2}}{4}}$

Touchy Circles

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