1993 AHSME Problems/Problem 8

Revision as of 20:58, 27 May 2021 by Logsobolev (talk | contribs) (Solution)

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 (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
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