Drawing the level curve of $x^2-y^2$ To do this, my professor solved for $x^2-y^2 = c$ when $c = 0$, $c > 0$ and $c < 0$. Why these particular values?
I know how to draw the curves for those values I don't understand why zero in particular.
 A: If you just solved the simple problem you'd see why the contours for $c=0$ are  "interesting":  They are straight lines (!!) amidst everywhere curved surfaces.

Fascinating!
What a terrible shame:  You ask others to solve this trivial problem for you and to reveal what is interesting rather than have the self-reliance and joy of discovering a little mathematical result on your own.  
Too bad.
A: If the equation had been $f(x, y) = x^2 - y^2 + 3$, then $f = 0$ would not have been an interesting level-set ... instead, $f = 3$ would have been interesting. Why is it interesting? Well, it's really because $x^2 - y^2$ factors into a product of linear terms, namely $x^2 - y^2 = (x-y )(x+y)$, and when $uv = 0$, one of $u$ or $v$ must be zero, so the plot is a pair of straight lines. 
This is more generally true: if you're hoping to plot level-sets for a polynomial $f$, you'd like to find values $c$ such that $f(x, y) - c$ actually factors, because those will generally be the "interesting" ones (the ones where the topology of the level sets change in some way, typically). 
As for this example, how did the prof know that $c = 0$ was special? Presumably she'd worked out the problem previously (or chose the values to make $c = 0$ special). That was (presumably) done as a way to set up this experience for you, so that you could make the leap to saying "Hey, what happened there? How did the left-and-right-pointing hyperbolas become up-and-down-pointing hyperbolas? What the heck is going on here?" Of course, that sort of experience and "aha! moment" only work for those willing to have the experience. 
