1990 AHSME Problems/Problem 6

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, $\fbox{D}$

1990 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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1990 AHSME Problems/Problem 6

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