Difference between revisions of "1990 AHSME Problems/Problem 6"

(Created page with "== Problem == Points <math>A</math> and <math>B</math> are <math>5</math> units apart. How many lines in a given plane containing <math>A</math> and <math>B</math> are <math>2</...")
 
 
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== Solution ==
 
== Solution ==
<math>\fbox{D}</math>
+
The lines have to be tangent to both of these circles.
 +
<asy>
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dot((0,0));dot((5,0));
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label("$A$",(0,0),S);label("$B$",(5,0),S);
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draw(Circle((0,0),2));draw(Circle((5,0),3));
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real m = sqrt(6)/12;
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path p = (-.4-3,4*sqrt(6)/5-3*m)--(4.4+4,6*sqrt(6)/5+4*m);
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draw(p,dotted);draw(reflect((0,0),(1,0))*p,dotted);draw((2,-4)--(2,4),dotted);
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</asy>
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By inspection, <math>\fbox{D}</math>
  
 
== See also ==
 
== See also ==

Latest revision as of 03:39, 4 February 2016

Problem

Points $A$ and $B$ are $5$ units apart. How many lines in a given plane containing $A$ and $B$ are $2$ units from $A$ and $3$ units from $B$?

$\text{(A) } 0\quad \text{(B) } 1\quad \text{(C) } 2\quad \text{(D) } 3\quad \text{(E) more than }3$

Solution

The lines have to be tangent to both of these circles. [asy] dot((0,0));dot((5,0)); label("$A$",(0,0),S);label("$B$",(5,0),S); draw(Circle((0,0),2));draw(Circle((5,0),3)); real m = sqrt(6)/12; path p = (-.4-3,4*sqrt(6)/5-3*m)--(4.4+4,6*sqrt(6)/5+4*m); draw(p,dotted);draw(reflect((0,0),(1,0))*p,dotted);draw((2,-4)--(2,4),dotted); [/asy] By inspection, $\fbox{D}$

See also

1990 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
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All AHSME Problems and Solutions

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