Solve this following


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


  1. parallel to $x$-axis

  2. with slope $\frac{2}{3}$

  3. with slope $\frac{3}{2}$

  4. parallel to $\mathrm{y}$-axis

Correct Option: , 2


Let the centroid of $\triangle \mathrm{PQR}$ is $(\mathrm{h}, \mathrm{k}) \& \mathrm{P}$ is $(\alpha, \beta)$, then

$\frac{\alpha+1+3}{3}=\mathrm{h} \quad$ and $\quad \frac{\beta+4-2}{3}=\mathrm{k}$

$\alpha=(3 h-4) \quad \beta=(3 k-4)$

Point $P(\alpha, \beta)$ lies on line $2 x-3 y+4=0$

$\therefore 2(3 h-4)-3(3 k-2)+4=0$

$\Rightarrow$ locus is $6 x-9 y+2=0$


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