Note: c = 0 gives the level curves x = 0 and y = 0 (which obviously pass through the origin).
Clearly this example is one of an infinite number of possibilities. I can only hope the OP has not been discouraged from contemplating it as a solution to his/her problem.
(I have switched the standard notation slightly [see red] to be consistent with the example I gave).
It defines a hyperbola, a parabola, an ellipse, a circle or a pair of lines depending on the value of the invariants of the curve.
A little bit of research should show origins of the example I gave.