2013 AMC 10B Problem 14

The condition is $x\clubsuit y=y\clubsuit x$, so $x^2y-xy^2=y^2x-yx^2$. Combining like terms gives $2x^2y-2xy^2=2xy(x-y)=0$. Thus $x=0$, $y=0$, or $x=y$. These are three lines. Thus, the correct answer is E .