In the above problem, coordinates of the point P such that |PA – PB| is minimum are

The minimum value of |PA – PB| is zero which is attained if PA = PB, i.e., P must lie on the perpendicular bisector of AB.
The equation of perpendicular bisector of AB is

$y=\frac{1}{2}=2(x-1) or y=2x-\frac{3}{2}$

Solving it with the given line, we get $P\equiv \left(-\frac{9}{20},-\frac{12}{5}\right)$