Consider the points $A=(0,1)\&B=(2,0),P$ be a point on the line $4x+3y+9=0$ Co-ordinates of the point ‘P’ such that $|PA-PB|$ is maximum is

$\left|AB\right|\ge |PA-PB|$

So $PA-PB$ is maximum when P is collinear with AB.
Equation of  $AB\Rightarrow y-1=\frac{-1}{2}(x-0)$
 $\Rightarrow x+2y=2\to \left(i\right)$
& solve with  $4x+3y+9=0\to \left(ii\right)$
(i) & (ii)
$P=(\frac{-24}{5},\frac{17}{5})$