This applet allows quick explorations of the general form of conic sections.
A x² + B x y + C y² + D x + E y + F = 0
If B² - 4A C < 0, the equation represents an ellipse. (If A = C and B = 0, the equation represents a circle, which is a special case of an ellipse.)
If B² - 4A C = 0, the equation represents a parabola.
If B² - 4A C > 0, the equation represents a hyperbola. (If we also have A + C = 0, the equation represents a rectangular hyperbola.)
B² - 4A C is called the discriminant of the general formula for conic sections (although is looks the same as the discriminant of the quadratic equation, it is used differently and the coefficients are not the same).
References: Conic_section and an old textbook page, Inspiration site: Mathlet-Conics
A B Cron, Created with GeoGebra