. . The general equation of a Conic Let S x y ) be the focus, l the directrix, and e be the eccentricity. Let P x y ) be the moving point.
By the definition of conic, we have SP PM = constant = e , ...( ) Where SP = and PM = perpendicular distance from P x y ( , ) to the line lx my = = lx my l m From ( ) we get SP = e PM ⇒ ( = e lx my l m . On simplification the above equation takes the form of general second-degree equation Ax Bxy Cy Dx Ey F , where e l l m B lme l m C e m l m Now , B AC l m e l m e l l m e m l m e yielding the following cases: (i) B AC = Û e = ⇔ the conic is a parabola, (ii) B AC < Û < < e ⇔ the conic is an ellipse, (iii) B AC > Û e > ⇔ the conic is a hyperbola.