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Question:
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 :
Correct Option: , 2
Solution:
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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