Given points P (2,3),Q (4,-2),R (a,0) what should be the value of a if |PR-RQ| Is maximum?

Question

Given points P (2,3),Q (4,-2),R (a,0) what should be the value of a if |PR-RQ| Is maximum ?

I tried that maybe the points are collinear but I'm getting wrong answer applying collinearity condition i.e a=16/5 but answer is 8.Help !

Answer 1

Let the point be $R(x,0)$.
Construct a function $f(x)=PR - RQ$ using the distance formula.

$$f(x ) =\sqrt{\left(2-x\right)^2+3^2}-\sqrt{\left(x-4\right)^2+2^2}$$

Find its stationary points and check for local maxima or minima.
Now check for which stationary point $|f(x)|$ is maximum.

Note: Luckily for your question, $f'(x)$ has only one stationary point, at $x= 8$.