The coordinates of the point P on the line $2x+3y+1=0$ , such that $\left|PA-PB\right|$ is maximum, where A is (2, 0) and B is (0, 2) is

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

Maximum value of $\left|PA-PB\right|is\left|AB\right|$ , which is possible only when P, A, B are collinear.

If $P\left(x,y\right)$ then equation AB is $\frac{x}{2}+\frac{y}{2}=1\Rightarrow x+y=2$    ….(i)

            Now solving Eq. (i) $and2x+3y+1=0$   ….(ii)

There we get,  $x=7,y=-5$

                 $\therefore P=(7,-5)$