Level curves (or contour curves) of a certain 3 dimensional function.

[Almost935]

I can't seem to quite comprehend the level curves for f(x,y)=xy. I realize this should be very simple yet the answer eludes me. f(x,y) for this the two dimensional representation of this function will be substituted with k(a fixed number). I would greatly appreciate somebody explaining my contour diagram to me and what the graph of k(fixed number)=xy even resembles in two dimensions and why.

 

[ShayanJ]

xy=k \Rightarrow y=\frac{k}{x}
k=0 \Rightarrow \left\{ \begin{array}{cc} y=0 \ \ \ x \ axis \\ x=0 \ \ \ y \ axis \end{array} \right.
k\neq 0 \Rightarrow hyperbolas with x and y axes as asymptotes.