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If the vertices \mathrm{\mathrm{A}\: and \: \mathrm{B}} of a triangle \mathrm{ABC} are given by \mathrm{(1,3), and (2,-7)} and \mathrm{C} moves along the line  \mathrm{L: 3 x+2 y+1=0,}the locus of the centroid of the triangle \mathrm{A B C} is a straight line parallel to
 

Option: 1

\mathrm{AB}

 


Option: 2

\mathrm{BC}
 


Option: 3

\mathrm{CA}
 


Option: 4

' \mathrm{L} '


Answers (1)

Let \mathrm{(h, k)} be the centroid of \mathrm{\triangle A B C} with \mathrm{ C} as \mathrm{(\alpha, \beta)}

\mathrm{ \therefore \quad h=\frac{1+2+\alpha}{3} \quad \Rightarrow \alpha=3 h-3 }

\mathrm{ \quad \therefore \quad k=\frac{3-7+\beta}{3} \quad \Rightarrow \beta=3 \mathrm{k}+4 }

\mathrm{ \text { since } \mathrm{C}(\alpha, \beta) \text { lies on } \mathrm{L} . }

\mathrm{ \therefore \quad 3(3 \mathrm{~h}-3)+2(3 \mathrm{k}+4)+1=0 }

\mathrm{ \Rightarrow 9 \mathrm{~h}+6 \mathrm{k}=0 }

Locus of  \mathrm{ (h, k)\: is \: 3 x+2 y=0 } which is parallel to \mathrm{ ' L '. }

Hence option 4 is correct.

Posted by

Kshitij

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