Question

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 $\Delta PQR$ is a line :

• A. parallel to x-axis
• B. parallel to y-axis
• C. with slope $\frac{3}{2}$
• D. with slope $\frac{2}{3}$

Answer 1

The correct option is D with slope $\frac{2}{3}$


Let the centroid of the $\Delta PQR$ is $C(\alpha ,\beta )$
$\therefore \alpha =\frac{h+1+3}{3}\Rightarrow h=3\alpha -4$
$\beta =\frac{4+k-2}{3}\Rightarrow k=3\beta -2$
$P(h,k)$ passes through the line $2x-3y+4=0$
$\therefore 2(3\alpha -4)-3(3\beta -2)+4=0$
$\Rightarrow 6\alpha -9\beta +2=0$
Hence, the locus is $6x-9y+2=0$
$\therefore$ slope $=\frac{6}{9}=\frac{2}{3}$