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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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