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</...")
 
 
Line 10: Line 10:
  
 
== Solution ==
 
== Solution ==
<math>\fbox{D}</math>
+
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, <math>\fbox{D}</math>
  
 
== See also ==
 
== See also ==

Latest revision as of 04:39, 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
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 26 27 28 29 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png