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Difference between revisions of "2010 AMC 12A Problems/Problem 13"

(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…')
 
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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.
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[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 $k$ do the graphs of $x^2+y^2=k^2$ and $xy = k$ not intersect?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

Solution

We can see that the function $xy = k$ is symmetric to the line $y=x$, and the distance to the origin approaches infinity as the function approaches either the $x$-axis or the $y$-axis. Therefore, assuming that graphs don't intersect, the point at which the function $xy = k$ is closest to the function $x^2+y^2=k^2$ (which is clearly a circle) is when $|x|=|y|$. It follows that at these points, the magnitude of the $x$ and $y$ values for the function $xy = k$ will be $\sqrt{|k|}$.

All of these points are found at angles $\frac{\pi}{4}$, $\frac{3\pi}{4}$, $\frac{5\pi}{4}$, or $\frac{7\pi}{4}$, so the minimum distance from the origin to the function $xy = k$ is $\sqrt{2}(\sqrt{|k|})$.

The distance from the circle to the origin is always $k$. 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 $-1$ and $1$, a total of $\boxed{2\ \textbf{(C)}}$ $k$ values.


[Images of the graphs of these functions would really help to understand and visualize this solution.]

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