A) \[2x+3y=9\]
B) \[2x-3y=7\]
C) \[3x+2y=5\]
D) \[3x-2y=3\]
Correct Answer: A
Solution :
[a] Let (x, y) be the coordinates of the vertex C and \[({{x}_{1}},{{y}_{1}})\]be the coordinates of the centroid of the triangle. Therefore, \[{{x}_{1}}=\frac{x+2-2}{3}\] \[{{y}_{1}}=\frac{y-3+1}{3}\] Or \[{{x}_{1}}=\frac{x}{3}\] And \[{{y}_{1}}=\frac{y-2}{3}\] The centroid lies on the line \[2x+3y=1\], so \[{{x}_{1}}\]and \[{{y}_{1}}\]satisfy the equation of the line. Thus \[2{{x}_{1}}+3{{y}_{1}}=1\] \[\Rightarrow 2\left( \frac{x}{3} \right)+3\left( \frac{y-2}{3} \right)=1\] \[\Rightarrow 2x+3y=9\] The above equation is the locus of the vertex c.
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