Y^2 - x^2 = 4 is it an ellipse, or a circle

This is actually a hiperbole. The circle's equation (which is a special case of ellipsis) is actually x^2 + y^2 = r^2. With your equation it would be like x^2 + y^2 = - 4, which would basically make the radius value negative, which is invalid (well, not invalid, but you get the meaning, I guess).