If $A(2,-3)$ and $B(-2,1)$ are two vertices of a triangle and third vertex moves on the line $2 x+3 y=9$, then the locus of the centroid of the triangle is

• A

  $2 x-3 y=1$
• B

  $x-y=1$
• Correct answer
  C

  $2 x+3 y=1$
• D

  $2 x+3 y=3$

Solution

Given $A(2,-3) B(-2,1)$ The third vertex lies on $2 x+3 y=9$ i.e. $C\left(x, \frac{9-2 x}{3}\right)$ $\therefore$ Let $P(h, k)$ be any point on the required locus i.e $P$ is the centroid of the triangle $A B C$ $$ \begin{aligned} & \Rightarrow\left(\frac{2-2+x}{3}, \frac{-3+1+\frac{9-2 x}{3}}{3}\right)=(h, k) \\ & \therefore h=\frac{x}{3}, k=\frac{3-2 x}{9} \end{aligned} $$ Eliminating $x$ from the above equations $$ \begin{aligned} & \Rightarrow 9 k=3-2(3 h) \\ & \Rightarrow 9 \mathrm{k}=3-6 h \\ & \Rightarrow 2 h+3 k=1 \end{aligned} $$ Hence, the locus of $P(h, k)$ is $2 x+3 y=1$

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