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  4. 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 ΔPQR is a line :

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Q. 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 :

JEE MainJEE Main 2019Straight Lines

Solution:

Let the centroid of $\Delta PQR$ is $( h , k ) \& P$ is $(\alpha, \beta)$, then
$\frac{\alpha+1+3}{3}= h \quad$ and $\quad \frac{\beta+4-2}{3}= k$
$\alpha=(3 h -4) \quad \beta=(3 k -4)$
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 $6 x-9 y+2=0$