Question

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

$0$

$1$

$2$

$4$

$8$

Answer: C

Concepts:circle hyperbola casework

Difficulty rating: 1590

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.

2010 AMC 12A Problem 13

Answer: C

Difficulty rating: 1590

Solution:

Problem 13 in Other Years