In the context of a parabola, what is a focus? What is a directrix? What are the advantages of expressing the equation of a parabola in focus-directrix form?
A parabola can be defined as a curve with a given point called the focus where every point on the curve is the same distance from the focus as the perpendicular distance to a given line called the directrix.
Many applications of parabolas involve receivers or transmitters that are formed from rotations of a parabola. Satellite dishes, headlights on cars, telescope mirrors, etc... are often paraboloids. We would need to be able to find the focus, as this is the point that incoming rays get sent to. (Or for transmitters like light reflectors, the point source is located at the focus and the outgoing rays leave in parallel lines.)
In focus-directrix form `x^2=4py` we know the distance from the vertex to the focus is p.