Finding Parabolas based solely on arbitrary point and line.

Question

Suppose one is given a point $(a,b)$ and a line $cx+dy+e=0$ in $\mathbb{R}^2$ such that $c$ and $d$ are not both equal to zero. How (if possible) would one give an equation of a parabola with focus as the given point and directrix the given line? I would argue it is clear how it should be done if $c=0$ or $d=0$, but what would change otherwise?

Answer 1

The definition of the parabola is:

A parabola is the geometrical locus of the points for which the distances from a point of coords $(a,b)$ and a line (the directrix of the parabola) are equal.

Let $P(x,y)$. By Pythagora's theorem, we know that the distance from $P$ to $A(a,b)$ is: $$\sqrt{(x-a)^2 +(y-b)^2}$$

And, by the distance formula, we have: $$\frac{cx+dy+e}{\sqrt{c^2+d^2}}$$

This two lenght have to be equal, so the equation becomes: $$\sqrt{(x-a)^2 +(y-b)^2}=\frac{cx+dy+e}{\sqrt{c^2+d^2}} \leftrightarrow (x-a)^2 +(y-b)^2=\frac{(cx+dy+e)^2}{c^2+d^2}$$