The General Equation of a Conic

Let $S(x_1,y_1)$ be the focus, $lx+my+n=0$ the directrix, $e$ the eccentricity, and $P(x,y)$ the moving point. By Definition 5.2, $\dfrac{SP}{PM}=e$, i.e. $SP^2=e^2PM^2$, where $SP=\sqrt{(x-x_1)^2+(y-y_1)^2}$ and $PM=\dfrac{|lx+my+n|}{\sqrt{l^2+m^2}}$ is the perpendicular distance from $P$ to the directrix. Substituting and simplifying always produces a second-degree equation

whose coefficients (worked out from $l,m,e$) satisfy

which reduces to the clean rule

$B^2-4AC=0 \iff e=1 \iff \text{parabola}, \qquad B^2-4AC<0 \iff 0<e<1 \iff \text{ellipse}, \qquad B^2-4AC>0 \iff e>1 \iff \text{hyperbola}.$ …