Difference between revisions of "2001 AMC 12 Problems/Problem 18"

Revision as of 19:02, 23 August 2023

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

$\begin{align*} s^2 + (1-r)^2 & = (1+r)^2 \\ (4-s)^2 + (4-r)^2 & = (4+r)^2 \end{align*}$

Simplifying both, we get

$\begin{align*} s^2 & = 4r \\ (4-s)^2 & = 16r \end{align*}$

As in our case both $r$ 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 $\frac{4-s}s=-2$ , 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. By coincidence, due to the $4:1$ ratio between radii of $A$ and $B$ , this circle turns out to have the same radius as circle $B$ , with center directly left of center $B$ , and tangent to $B$ directly above center $A$ .) 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 $a$ and $b$ , the radius $c$ of the small circle is $c = \frac{ab}{\left(\sqrt{a}+\sqrt{b}\right)^2} = \frac{1}{\left(1/\sqrt{a}+1/\sqrt{b}\right)^2}$ .

Equivalently, we have that $1/\sqrt{c} = 1/\sqrt{a} + 1/\sqrt{b}$ .

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

@@ Line 87: / Line 87: @@
 As in our case both <math>r</math> and <math>s</math> are positive, we can divide the second one by the first one to get <math>\left( \frac{4-s}s \right)^2 = 4</math>.
-Now there are two possibilities: either <math>\frac{4-s}s=-2</math>, or <math>\frac{4-s}s=2</math>. In the first case clearly <math>s<0</math>, 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 <math>A</math>.) The second case solves to <math>s=\frac 43</math>. We then have <math>4r = s^2 = \frac {16}9</math>, hence <math>r = \boxed{\frac 49}</math>.
+Now there are two possibilities: either <math>\frac{4-s}s=-2</math>, or <math>\frac{4-s}s=2</math>. In the first case clearly <math>s<0</math>, 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. By coincidence, due to the <math>4:1</math> ratio between radii of <math>A</math> and <math>B</math>, this circle turns out to have the same radius as circle <math>B</math>, with center directly left of center <math>B</math>, and tangent to <math>B</math> directly above center <math>A</math>.) The second case solves to <math>s=\frac 43</math>. We then have <math>4r = s^2 = \frac {16}9</math>, hence <math>r = \boxed{\frac 49}</math>.
-More generally, for two large circles of radius <math>a</math> and <math>b</math>, the radius <math>c</math> of the small circle is <math>c = \frac{ab}{\left(\sqrt{a}+\sqrt{b}\right)^2} = \frac{1}{\left(1/\sqrt{a}+1/\sqrt{b}\right)^2}</math>. Equivalentlyx we have that <math>1/\sqrt{c} = 1/\sqrt{a} + 1/\sqrt{b}</math>.
+More generally, for two large circles of radius <math>a</math> and <math>b</math>, the radius <math>c</math> of the small circle is <math>c = \frac{ab}{\left(\sqrt{a}+\sqrt{b}\right)^2} = \frac{1}{\left(1/\sqrt{a}+1/\sqrt{b}\right)^2}</math>.
+Equivalently, we have that <math>1/\sqrt{c} = 1/\sqrt{a} + 1/\sqrt{b}</math>.
 === Solution 2 ===

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