Difference between revisions of "2005 AIME II Problems/Problem 15"
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== Problem == | == Problem == | ||
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Let <math> w_1 </math> and <math> w_2 </math> denote the circles <math> x^2+y^2+10x-24y-87=0 </math> and <math> x^2 +y^2-10x-24y+153=0, </math> respectively. Let <math> m </math> be the smallest possible value of <math> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally tangent to <math> w_2 </math> and internally tangent to <math> w_1. </math> Given that <math> m^2=\frac pq, </math> where <math> p </math> and <math> q </math> are relatively prime integers, find <math> p+q. </math> | Let <math> w_1 </math> and <math> w_2 </math> denote the circles <math> x^2+y^2+10x-24y-87=0 </math> and <math> x^2 +y^2-10x-24y+153=0, </math> respectively. Let <math> m </math> be the smallest possible value of <math> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally tangent to <math> w_2 </math> and internally tangent to <math> w_1. </math> Given that <math> m^2=\frac pq, </math> where <math> p </math> and <math> q </math> are relatively prime integers, find <math> p+q. </math> | ||
== Solution == | == Solution == | ||
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{{solution}} | {{solution}} | ||
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== See also == | == See also == | ||
| − | *[[2005 AIME II Problems/Problem 14| Previous problem]] | + | * [[2005 AIME II Problems/Problem 14| Previous problem]] |
* [[2005 AIME II Problems]] | * [[2005 AIME II Problems]] | ||
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| + | [[Category:Intermediate Geometry Problems]] | ||
Revision as of 22:35, 7 September 2006
Problem
Let 
and 
denote the circles 
and 
respectively. Let 
be the smallest possible value of 
for which the line 
contains the center of a circle that is externally tangent to and internally tangent to 
Given that 
where 
and 
are relatively prime integers, find 
Solution
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