For example, the quadratic equation xy = 1 gives a hyperbola, while x2 + y2 = 4 is a circle and x2 + 2y2 = 4 is an ellipse. Even a quadratic as beastly as x2 + 2xy + y2 + x + 3y = 2 has to be one of the four possibilities above. It turns out to be a parabola.
