2010 AMC 12A Problem 13

Below is the professionally curated solution for Problem 13 of the 2010 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 12A solutions, or check the answer key.

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Concepts:circle hyperbola casework

Difficulty rating: 1590

13.

For how many integer values of $k$ do the graphs of $x^2+y^2=k^2$ and $xy=k$ not intersect?

$0$

$1$

$2$

$4$

$8$

Solution:

For $k=0,$ the graph of $x^2+y^2=0$ is the single point $(0,0)$ and $xy=0$ is the two axes, which meet at the origin, so the graphs intersect.

For $k\ne0,$ the circle has radius $|k|,$ and the hyperbola $xy=k$ has its two vertices nearest the origin at distance $\sqrt{2|k|}.$ The graphs meet exactly when $|k|\ge\sqrt{2|k|},$ that is $|k|\ge2.$

So they fail to intersect only when $|k|=1,$ namely $k=1$ and $k=-1,$ giving $2$ values.

Thus, C is the correct answer.

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