Problem:
On a sheet of paper, Isabella draws a circle of radius 2 , a circle of radius 3 , and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly lines. How many different values of are possible?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
If the smaller circle is in the interior of the larger circle, there are no common tangent lines. If the smaller circle is internally tangent to the larger circle, there is exactly one common tangent line. If the circles intersect at two points, there are exactly two common tangent lines. If the circles are externally tangent, there are exactly three tangent lines. Finally, if the circles do not intersect, there are exactly four tangent lines. Therefore, can be any of the numbers , or 4 , which gives possibilities.
The problems on this page are the property of the MAA's American Mathematics Competitions