question_answer

Let \[A(2,-3)\]and \[B(-2,1)\]be vertices of a triangle ABC. If the centroid of this triangle moves on the line \[2x+3y=1\], then the locus of the vertex C is the line [AIEEE 2004]

A) \[3x-2y=3\]

B) \[2x-3y=7\]

C) \[3x+2y=5\]

D) \[2x+3y=9\]

Correct Answer: D

Solution :

Let the third vertex be \[({{x}_{1}},\,{{y}_{1}})\] then Centroid\[(G)\equiv \left( \frac{{{x}_{1}}+2-2}{3},\,\frac{{{y}_{1}}-3+1}{3} \right)\] i.e., \[G\,\left( \frac{{{x}_{1}}}{3},\,\frac{{{y}_{1}}-2}{3} \right)\] Given, centroid of triangle moves on the line   \[2x+3y=1\] \[\therefore \] \[2\,\left( \frac{{{x}_{1}}}{3} \right)+3\left( \frac{{{y}_{1}}-2}{3} \right)=1\] i.e., \[2{{x}_{1}}+3{{y}_{1}}=9\] Locus of \[({{x}_{1}},\,{{y}_{1}})\] is \[2x+3y=9\].