Triangle inequality in coordinate geometry

Question

Let $P,Q,R$ denote points $(2,3),(4,-2),(h,0)$ respectively.

Find value of $h$ if $|PR-RQ|$ is maximum.

My Attempt: I am trying to use triangle inequality i.e.

$|PR-RQ|\leq PQ$

but not getting the answer.

Answer 1

You can draw a picture, so we can see that we can not have $|PR-QR|=PQ$ because $Q$ can not be between $P$ and $R$ or $P$ can not be between $Q$ and $R$.

The motivation of taking the reflection of $Q$, namely $Q'$, is that we can have $Q'$ is between $P$ and $R$, and then we can use triangle inequality and the equality can be happen

Another way: Consider the function $f(h)=\sqrt{h^2-4h+13}-\sqrt{h^2-8h+20}$, we can find that $f'(h)=0$ when $h=8$ or $h=16/5$, then max$|f(h)|$=$\sqrt{5}$ at $h=8$

https://www.wolframalpha.com/input/?i=sqrt(x%5E2-4x%2B13)-sqrt(x%5E2-8x%2B20)