My question is on deriving the parabola equation given a directrix and a focus. I think below is a valid equation for a parabola? So, how do you avoid extraneous solutions here: $$\sqrt{(x - f_x)^2 + (y - f_y)^2} = \sqrt{(y + f_y)^2} $$
It seems in the sources that I've seen that the next step to simplifying this equation is to square both sides and create the following like equation: $$ (x - f_x)^2 + (y - f_y)^2 = (y + f_y)^2 $$ Is this a valid step? It seems so, but why aren't there extraneous solutions? I get that solutions to the original equation are solutions to the second derived equation but I don't get why there wouldn't be extra solutions for the second equation that are not solutions to the first equation?
Squaring both sides of an equation can, but doesn't necessarily, introduce extraneous solutions. In particular, they can happen if the two sides can have different signs.
It's true that
$$ L = R \implies L^2 = R^2 $$
but not true that
$$ L^2 = R^2 \implies L = R $$
since we also have the possibility that $L = -R \neq 0$ and $L^2 = R^2$. Solutions to $L = -R \neq 0$, if any, are the extraneous solutions to $L^2 = R^2$ which are not solutions to the original $L=R$.
In this case, since the square root of a real number is never negative, it's not possible that
$$ \sqrt{(x-f_x)^2+(y-f_y)^2} = -\sqrt{(y+f(y))^2} \neq 0 $$
So this time, no extraneous solutions exist.