Difference between revisions of "1993 AHSME Problems/Problem 8"

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== Solution ==
 
== Solution ==
There are two radius 3 circles to which <math>C_1</math> and <math>C_2</math> are both externally tangent.  One touches the tops of <math>C_1</math> and <math>C_2</math> and extends upward, and the other the other touches the bottoms and extends downward.  There are also two radius 3 circles to which <math>C_1</math> and <math>C_2</math> are both internally tangent, one touching the tops and encircling downward, and the other touching the bottoms and encircling upward.  There are two radius 3 circles passing through the point where <math>C_1</math> and <math>C_2</math> are tangent.  For one <math>C_1</math> is internally tangent and <math>C_2</math> is externally tangent, and for the other <math>C_2</math> is externally tangent and <math>C_1</math> is internally tangent.
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There are two radius 3 circles to which <math>C_1</math> and <math>C_2</math> are both externally tangent.  One touches the tops of <math>C_1</math> and <math>C_2</math> and extends upward, and the other the other touches the bottoms and extends downward.  There are also two radius 3 circles to which <math>C_1</math> and <math>C_2</math> are both internally tangent, one touching the tops and encircling downward, and the other touching the bottoms and encircling upward.  There are two radius 3 circles passing through the point where <math>C_1</math> and <math>C_2</math> are tangent.  For one <math>C_1</math> is internally tangent and <math>C_2</math> is externally tangent, and for the other <math>C_1</math> is externally tangent and <math>C_2</math> is internally tangent.
  
 
<math>\fbox{D}</math>
 
<math>\fbox{D}</math>

Latest revision as of 20:59, 27 May 2021

Problem

Let $C_1$ and $C_2$ be circles of radius 1 that are in the same plane and tangent to each other. How many circles of radius 3 are in this plane and tangent to both $C_1$ and $C_2$?

$\text{(A) } 2\quad \text{(B) } 4\quad \text{(C) } 5\quad \text{(D) } 6\quad \text{(E) } 8$

Solution

There are two radius 3 circles to which $C_1$ and $C_2$ are both externally tangent. One touches the tops of $C_1$ and $C_2$ and extends upward, and the other the other touches the bottoms and extends downward. There are also two radius 3 circles to which $C_1$ and $C_2$ are both internally tangent, one touching the tops and encircling downward, and the other touching the bottoms and encircling upward. There are two radius 3 circles passing through the point where $C_1$ and $C_2$ are tangent. For one $C_1$ is internally tangent and $C_2$ is externally tangent, and for the other $C_1$ is externally tangent and $C_2$ is internally tangent.

$\fbox{D}$

See also

1993 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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