2010 AMC 12A Problem 13

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.