2010 AMC 12A Problems/Problem 13: Difference between revisions
Created page with '== Problem 13 == For how many integer values of <math>k</math> do the graphs of <math>x^2+y^2=k^2</math> and <math>xy = k</math> not intersect? <math>\textbf{(A)}\ 0 \qquad \tex…' |
|||
| Line 16: | Line 16: | ||
It is then easy to see that the only values that satisfy the inequality are <math>-1</math> and <math>1</math>, a total of <math>\boxed{2\ \textbf{(C)}}</math> <math>k</math> values. | It is then easy to see that the only values that satisfy the inequality are <math>-1</math> and <math>1</math>, a total of <math>\boxed{2\ \textbf{(C)}}</math> <math>k</math> values. | ||
[Images of the graphs of these functions would really help to understand and visualize this solution.] | |||
Revision as of 20:09, 10 February 2010
Problem 13
For how many integer values of 
do the graphs of 
and 
not intersect?
Solution
We can see that the function is symmetric to the line 
, and the distance to the origin approaches infinity as the function approaches either the 
-axis or the 
-axis. Therefore, assuming that graphs don't intersect, the point at which the function is closest to the function (which is clearly a circle) is when 
. It follows that at these points, the magnitude of the and values for the function will be 
.
All of these points are found at angles 
, 
, 
, or 
, so the minimum distance from the origin to the function is 
.
The distance from the circle to the origin is always . Therefore, we want to find all integer values such that
$|k| < \sqrt{|2k|}}$ (Error compiling LaTeX. Unknown error_msg)
It is then easy to see that the only values that satisfy the inequality are 
and 
, a total of 
values.
[Images of the graphs of these functions would really help to understand and visualize this solution.]
