Difference between revisions of "2006 AMC 12A Problems/Problem 19"
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| − | [[Circle]]s with [[center]]s <math>(2,4)</math> and <math>(14,9)</math> have [[radius | radii]] <math>4</math> and <math>9</math>, respectively. The equation of a common external [[tangent line|tangent]] to the circles can be written in the form <math>y=mx+b</math> with <math>m>0</math>. What is <math>b</math>? | + | [[Circle]]s with [[center_(geometry) | center]]s <math>(2,4)</math> and <math>(14,9)</math> have [[radius | radii]] <math>4</math> and <math>9</math>, respectively. The equation of a common external [[tangent line|tangent]] to the circles can be written in the form <math>y=mx+b</math> with <math>m>0</math>. What is <math>b</math>? |
<center>[[Image:AMC12_2006A_19.png]]</center> | <center>[[Image:AMC12_2006A_19.png]]</center> | ||
Revision as of 21:52, 29 December 2021
Problem
Circles with centers 
and 
have radii 
and 
, respectively. The equation of a common external tangent to the circles can be written in the form 
with 
. What is 
?
Solution
Let 
be the line that goes through and , and let 
be the line . If we let 
be the measure of the acute angle formed by and the x-axis, then 
. clearly bisects the angle formed by and the x-axis, so 
. We also know that and intersect at a point on the x-axis. The equation of is 
, so the coordinate of this point is 
. Hence the equation of is 
, so 
, and our answer choice is 
.
See also
| 2006 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 18 |
Followed by Problem 20 |
| 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 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
