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

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== Solution ==
 
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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_2</math> is externally tangent and <math>C_1</math> is internally tangent.
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<math>\fbox{D}</math>
 
<math>\fbox{D}</math>
  

Revision as of 19:58, 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 $C_2$ is externally tangent and $C_1$ is internally tangent.

$\fbox{D}$

See also

1993 AHSME (Problems • Answer Key • Resources)
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
All AHSME Problems and Solutions

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