Let $P(2,-3), Q(-2,1)$ be the vertices of the $\Delta P Q R .$ If the centroid of $\Delta P Q R$ lies on the line $2 x+3 y=1,$ then the locus of $R$ is

• Correct answer
  A

  $2 x+3 y=9$
• B

  $2 x-3 y=7$
• C

  $3 x+2 y=5$
• D

  $3 x-2 y=5$

Solution

Let the vertices of $R$ be $(h, k)$ $\therefore$ Centroid of triangle is given by $$ \left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}\right) $$ Le, $\left(\frac{2+h-2}{3}, \frac{-3+1+k}{3}\right)$ Le, $\left(\frac{h}{3}, \frac{-2+k}{3}\right)$ since, the point $\left(\frac{h}{3}, \frac{-2+k}{3}\right)$ lies on the line $2 x+3 y=1,$ therefore the point $\left(\frac{h}{3}, \frac{-2+k}{3}\right)$ must be satisfy the equation of the given line. Le., $2\left(\frac{h}{3}\right)+3\left(\frac{-2+k}{3}\right)=1$ $\Rightarrow \quad \frac{2 h-6+3 k}{3}=1$ $\Rightarrow \quad 2 h-6+3 k=3$ $\Rightarrow h+3 k=9$ Replace $h=x$ and $k=y,$ we get the required locus of $R$. ie, $\quad 2 x+3 y=9$

Hence, the correct answer is option: 'A'.