The general equation of a Conic
Let S ( x1 , y1 ) 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
and PM = perpendicular distance from P(x, y) to the line lx + my + n = 0
On simplification the above equation takes the form of general second-degree equation
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 , where
yielding the following cases:
(i) B2 - 4 AC = 0 Û e = 1 Û the conic is a parabola,
(ii) B2 - 4 AC < 0 Û 0 < e <1 Û the conic is an ellipse,
(iii) B2 - 4 AC > 0 Û e >1 Û the conic is a hyperbola.