Difference between revisions of "2006 AMC 12A Problems/Problem 19"
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== Problem == | == Problem == | ||
+ | [[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> | ||
− | {{ | + | <math> \mathrm{(A) \ } \frac{908}{119}\qquad \mathrm{(B) \ } \frac{909}{119}\qquad \mathrm{(C) \ } \frac{130}{17}\qquad \mathrm{(D) \ } \frac{911}{119}\qquad \mathrm{(E) \ } \frac{912}{119}</math> |
− | + | == Solution == | |
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− | == | + | Let <math>L_1</math> be the line that goes through <math>(2,4)</math> and <math>(14,9)</math>, and let <math>L_2</math> be the line <math>y=mx+b</math>. If we let <math>\theta</math> be the measure of the acute angle formed by <math>L_1</math> and the x-axis, then <math>\tan\theta=\frac{5}{12}</math>. <math>L_1</math> clearly bisects the angle formed by <math>L_2</math> and the x-axis, so <math>m=\tan{2\theta}=\frac{2\tan\theta}{1-\tan^2{\theta}}=\frac{120}{119}</math>. We also know that <math>L_1</math> and <math>L_2</math> intersect at a point on the x-axis. The equation of <math>L_1</math> is <math>y=\frac{5}{12}x+\frac{19}{6}</math>, so the coordinate of this point is <math>\left(-\frac{38}{5},0\right)</math>. Hence the equation of <math>L_2</math> is <math>y=\frac{120}{119}x+\frac{912}{119}</math>, so <math>b=\frac{912}{119}</math>, and our answer choice is <math>\boxed{\mathrm{E}}</math>. |
== See also == | == See also == | ||
− | + | {{AMC12 box|year=2006|ab=A|num-b=18|num-a=20}} | |
− | + | {{MAA Notice}} | |
[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] |
Latest revision as of 22:52, 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 |
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