Difference between revisions of "1998 AHSME Problems/Problem 23"
(New page: == Problem == The graphs of <math>x^2 + y^2 = 4 + 12x + 6y</math> and <math>x^2 + y^2 = k + 4x + 12y</math> intersect when <math>k</math> satisfies <math>a \le k \le b</math>, and for no o...) |
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== See also == | == See also == | ||
{{AHSME box|year=1998|num-b=22|num-a=24}} | {{AHSME box|year=1998|num-b=22|num-a=24}} | ||
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Revision as of 13:30, 5 July 2013
Problem
The graphs of 
and 
intersect when 
satisfies 
, and for no other values of . Find 
.
Solution
Both sets of points are quite obviously circles. To show this, we can rewrite each of them in the form 
.
The first curve becomes 
, which is a circle centered at 
with radius 
.
The second curve becomes 
, which is a circle centered at 
with radius 
.
The distance between the two centers is 
, and therefore the two circles intersect iff 
.
From 
we get that 
. From 
we get 
.
Therefore 
.
See also
| 1998 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 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. 
