Originally Posted by
Idea Let
$\displaystyle P=(a,b)$ and $\displaystyle Q=(c,d)$
$\displaystyle \overrightarrow{\text{AQ}}=\pm 2 \overrightarrow{\text{AP}}$
Translating into coordinates gives 2 linear equations
$\displaystyle \pm 2(a-1)=c-1$
$\displaystyle \pm 2(b-2)=d-2$
Two more linear equations:
$\displaystyle 2b=3a-5$
$\displaystyle c+d=12$
EDIT: Looking at the sketch provided by the OP, APQ doesn't look at all like a straight line in which case the problem has infinitely many solutions. Otherwise, only two solutions