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  4. 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

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Q. 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

J & K CETJ & K CET 2014Straight Lines

Solution:

Let coordinate of third vertex be $ C({{x}_{1}},\,{{y}_{1}}) $ Then, centroid of triangle
$=\left( \frac{2+(-2)+{{x}_{1}}}{3},\frac{-3+1+{{y}_{1}}}{3} \right)=\left( \frac{{{x}_{1}}}{3},\frac{{{y}_{1}}-2}{3} \right) $ Since, centroid of this 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 $
$ \Rightarrow $ $ 2{{x}_{1}}+3{{y}_{1}}-6=3 $
$ \Rightarrow $ $ 2{{x}_{1}}+3{{y}_{1}}=9 $
Hence, the locus of the vertex C is the line $ 2x+3y=9 $