2001 AMC 12 Problems/Problem 18

Problem

A circle centered at $A$ with a radius of 1 and a circle centered at $B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. What is the radius of the third circle?

$[asy] unitsize(0.75cm); pair A=(0,1), B=(4,4); dot(A); dot(B); draw( circle(A,1) ); draw( circle(B,4) ); draw( (-1.5,0)--(8.5,0) ); draw( A -- (A+(-1,0)) ); label("$1$", A -- (A+(-1,0)), N ); draw( B -- (B+(4,0)) ); label("$4$", B -- (B+(4,0)), N ); label("$A$",A,E); label("$B$",B,W); filldraw( circle( (12/9,4/9), 4/9 ), lightgray, black ); dot( (12/9,4/9) ); [/asy]$

$\text{(A) }\frac {1}{3} \qquad \text{(B) }\frac {2}{5} \qquad \text{(C) }\frac {5}{12} \qquad \text{(D) }\frac {4}{9} \qquad \text{(E) }\frac {1}{2}$

Solution

Solution 1

$[asy] unitsize(1cm); pair A=(0,1), B=(4,4), C=(4,1), S=(12/9,4/9); dot(A); dot(B); draw( circle(A,1) ); draw( circle(B,4) ); draw( (-1.5,0)--(8.5,0) ); draw( (A+(4,0)) -- A -- (A+(0,-1)) ); draw( A -- B -- (B+(0,-4)) ); label("$A$",A,N); label("$B$",B,N); label("$C$",C,E); label("$S$",S,N); filldraw( circle(S,4/9), lightgray, black ); dot(S); draw( rightanglemark(A,C,B) ); draw( S -- A ); draw( S -- B ); [/asy]$

In the triangle $ABC$ we have $AB = 1+4 = 5$ and $BC=4-1 = 3$ , thus by the Pythagorean theorem we have $AC=4$ .

Let $r$ be the radius of the small circle, and let $s$ be the perpendicular distance from $A$ to $\overline{BC}. Moreover, the small circle is tangent to both other circles, hence we have$ SA=1+r $and$ SB=4+r$.

We have$ (Error compiling LaTeX. Unknown error_msg)SA = \sqrt{s^2 + (1-r)^2} $and$ SB=\sqrt{(4-s)^2 + (4-r)^2}$. Hence we get the following two equations:

<cmath> \begin{align*} s^2 + (1-r)^2 & = (1+r)^2 \\ (4-s)^2 + (4-r)^2 & = (4+r)^2 \end{align*} </cmath>

Simplifying both, we get

<cmath> \begin{align*} s^2 & = 4r \\ (4-s)^2 & = 16r \end{align*} </cmath>

As in our case both$ (Error compiling LaTeX. Unknown error_msg)r $and$ s $are positive, we can divide the second one by the first one to get$ \left( \frac{4-s}s \right)^2 = 4$.

Now there are two possibilities: either$ (Error compiling LaTeX. Unknown error_msg)\frac{4-s}s=-2 $, or$ \frac{4-s}s=2 $. In the first case clearly$ s<0 $, hence this is not the correct case. (Note: This case corresponds to the other circle that is tangent to both given circles and the common tangent line - a large circle whose center is somewhere to the left of$ A $.) The second case solves to$ s=\frac 43 $. We then have$ 4r = s^2 = \frac {16}9 $, hence$ r = \boxed{\frac 49}$.

More generally, for two large circles of radius$ (Error compiling LaTeX. Unknown error_msg)a $and$ b $, the radius of the small circle is$ \frac{ab}{(a+b)^2}$

Solution 2

The horizontal line is the equivalent of a circle of curvature $0$ , thus we can apply Descartes' Circle Formula.

The four circles have curvatures $0, 1, \frac 14$ , and $\frac 1r$ .

We have $2\left(0^2+1^2+\frac {1}{4^2}+\frac{1}{r^2}\right)=\left(0+1+\frac 14+\frac 1r\right)^2$

Simplifying, we get $\frac{34}{16}+\frac{2}{r^2}=\frac{25}{16}+\frac{5}{2r}+\frac{1}{r^2}$

$\frac{1}{r^2}-\frac{5}{2r}+\frac{9}{16}=0$ $\frac{16}{r^2}-\frac{40}{r}+9=0$ $\left(\frac{4}{r}-9\right)\left(\frac{4}{r}-1\right)=0$

Obviously $r$ cannot equal $4$ , therefore $r = \boxed{\frac 49}$ .

Video Solution

https://youtu.be/zOwYoFOUg2U

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
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