5.0 General equation of Parabola

Let $S(a,b)$ be the focus and $lx+my+n=0$ is the equation of directrix. Let $P(x,y)$ be any point on parabola. Then,

$$SP=PM$$

$$\sqrt {{{\left( {x - a} \right)}^2} + {{\left( {y - b} \right)}^2}} = \frac{{\left| {lx + my + n} \right|}}{{\sqrt {{l^2} + {m^2}} }}$$

Squaring both sides, we get $${\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = \frac{{{{\left| {lx + my + n} \right|}^2}}}{{{l^2} + {m^2}}}$$ or,

$${m^2}{x^2} + {l^2}{y^2} - 2lmxy + x\ term + y\ term + constant\ term = 0$$

It is of form ${\left( {mx - ly} \right)^2} + 2gx + 2fy + c = 0$