A point P moves on the line $2x-3y+4=0$. If$Q (1,4) and R(3,-2)$ are fixed points, then the locus of the centroid of triangle PQR is a line 

Let G (h, k) be the centroid of $△\mathrm{PQR} \mathrm{and} \mathrm{P}=(\mathrm{\alpha },\mathrm{\beta })$
Then (h,k)=$\left(\frac{\mathrm{\alpha }+1+3}{3},\frac{\mathrm{\beta }+4-2}{3}\right)\Rightarrow \mathrm{h}=\frac{\mathrm{\alpha }+4}{3},\mathrm{k}=\frac{\mathrm{\beta }+2}{3}\Rightarrow \mathrm{\alpha }=3\mathrm{h}-4,\mathrm{\beta }=3\mathrm{k}-2$
P lies on 2x - 3y + 4 = 0 = 2α - 3ß + 4 = 0 $\Rightarrow$ 2(3h - 4) - 3(3k - 2) + 4 = 0 $\Rightarrow$ 6h - 9k + 2 = 0
$\therefore$Locus of G (h,k) is 6x -9y +2= 0 which is a line having slope $\frac{6}{9}=\frac{2}{3}$