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Let $A(2,3)$ and $B(-2,3)$ be vertices of a triangle ABC. If the centriod of this triangle moves on the line $2x+3y=1$ then the locus of the vertex C is the line .

$\begin{array}{1 1}(A)\;3x-2y=3\\(B)\;2x-3y=7\\(C)\;3x+2y=5\\(D)\;2x+3y=9\end{array}$

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A)
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Let the vertex $C$ be $(h,K)$ then the centroid of $\Delta ABC$ is $\bigg( \large\frac{2-2+h}{3},\frac{-3+1+k}{3}\bigg)$ or $\bigg(\large\frac{h}{3}, \frac{-2+k}{3}\bigg)$
It lies on $2x+3y=1$
=> $\large\frac{2h}{3}$$=2+k=1$
=> $2h+3k=9$
=> Locus of C is $2x+3y=9$
Hence D is the correct answer.
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