Are there any practical applications of the directrix of a parabola?

Question

I know of many applications for the focus of a parabola (satellite dish, whispering gallery, etc.), but haven't been able to find any for the directrix. An internet search has come up empty. I have interviewed several math teachers and none of them could help either. Any ideas?

Answer 1

This might not be what you're looking for, but here is a problem I've seen in a Calculus class. It doesn't really have a real-world application (that I know of), but the directrix does come up in an interesting way.

Let $S$ be the square with vertices $(-1,-1), (1,-1), (1,1)$ and $(-1,1)$ (the square of side length 2 centered at the origin), and let $A$ be the set of all points closer to the origin than to the sides of $S$. Find the area of $A$.

The boundary of $A$ is the set of all points whose distance to the origin equals the distance to one of the sides of the square. Essentially $A$ is the region bounded by the four parabolas with focus at the origin and a side of the square as directrix.

Answer 2

You can always make up a real-world application...

Say you are walking along a parabola-shaped path, when you see a bear at the focus. If all other bears are on the other side of the directrix, then you are safe. But if there is a second bear on this side of the directrix, then watch out, because then your path will sometimes be nearer one bear and sometimes nearer the other, and when you get near this territorial boundary the bears will charge towards you to maintain their claim to you as food.

If you want to simulate a "point" reflector, that reflects waves aimed at the point directly back to their source, then might think you could use a mirrored ball centered at the point. But this has the problem that the waves arrive back at the source before they should. To solve this, you can instead use the outside of a parabola together with a mirror on the directrix.

Say there is a parabola-shaped lake, and you want to build two perpendicular roads, neither of which is blocked by the lake. Where can the crossroads be? Answer: Anywhere on the far side of the directrix.

By the way, the two applications you mention for the focus are actually the same, since satellites are so far away that their faint signal needs to be collected in the same way as a whisper.