Minimizing the distance between two points and a line

[Mr Davis 97]

Homework Statement

Let $A$ and $B$ be two given points, $l$ a given line. Find the point $P$ on $l$ such that $|AP-PB|$ is a maximum.

Homework Equations

The Attempt at a Solution

Suppose that the line $l$ and the line that goes through $A$ and $B$ (call it $\ell_{AB}$) are not parallel. Then our desired point $P$ is the the intersection of $l$ and $\ell_{AB}$. Is this correct? I'm pretty sure it is.

But what about when $l$ and $\ell_{AB}$ are parallel? There doesn't seem to be a maximum, since if you move a point $P$ along any direction of the line we'll just get greater and greater values for $|AP-PB|$

 

[RPinPA]

I think your intuition is correct, in which case you'd just note that in the parallel case $|AP - PB|$ is unbounded above.

But you should solve this analytically, maximizing by the usual methods. One hint: When working with distances, it will usually make your life a heck of a lot easier to work with the squared distance. rather than the absolute value. It's differentiable everywhere (no corner points) and the form of the derivatives is a lot simpler.

So set up the expression for $|AP - PB|^2$ and apply the standard methods.

Edit: I misread this problem. My general comments about using distance squared and avoiding square roots still stands, but at first glance I'm not sure if it's applicable to this problem. The general idea is to get a nicer function which is minimized / maximized when your original function is minimized / maximized, but simply squaring isn't going to do it here.

More Edit: My hunch is that the "nicer function" would be something involving trig functions, perhaps invoking the law of cosines in some way. Indeed, it may be the triangles involved that led you to your intuition on the nature of the solution.