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9.Straight Line
hard
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
with slope $\frac{3}{2}$
B
parallel to $x-$ axis
C
with slope $\frac{2}{3}$
D
parallel to $y-$ axis
(JEE MAIN-2019)
Solution
Let point $P$ is $\left( {\alpha ,\beta } \right)\,$ and center of $\Delta PQR$ is $(h,k)$, then $3h = \alpha + 1 + 3\,\,$ and $3k = \beta + 4 – 2$
$ \Rightarrow \alpha = 3h – 4$ and $\beta = 3k – 2$
Because $\left( {\alpha ,\beta } \right)$ lies on $2x-3y+4=0$
$ \Rightarrow 2\left( {3h – 4} \right) – 3\left( {k – 2} \right) + 4 = 0$
$ \Rightarrow $ losus is $6x-9y+2=0$ whose slope is $\frac{2}{3}$
Standard 11
Mathematics
