JEE MAIN - Mathematics (2004 - No. 42)
Let $$A\left( {2, - 3} \right)$$ and $$B\left( {-2, 1} \right)$$ 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 :
$$3x - 2y = 3$$
$$2x - 3y = 7$$
$$3x + 2y = 5$$
$$2x + 3y = 9$$
Explanation
Let the vertex $$C$$ be $$(h,k),$$ then the
centroid of $$\Delta ABC$$ is $$\left( {{{2 + (- 2) + h} \over 3},{{ - 3 + 1 + k} \over 3}} \right)$$
or $$\left( {{h \over 3},{{ - 2 + k} \over 3}} \right).$$ It lies on $$2x+3y=1$$
$$ \Rightarrow {{2h} \over 3} - 2 + k = 1$$
$$ \Rightarrow 2h + 3k = 9$$
$$ \therefore $$ Locus of $$C$$ is $$2x+3y=9$$
centroid of $$\Delta ABC$$ is $$\left( {{{2 + (- 2) + h} \over 3},{{ - 3 + 1 + k} \over 3}} \right)$$
or $$\left( {{h \over 3},{{ - 2 + k} \over 3}} \right).$$ It lies on $$2x+3y=1$$
$$ \Rightarrow {{2h} \over 3} - 2 + k = 1$$
$$ \Rightarrow 2h + 3k = 9$$
$$ \therefore $$ Locus of $$C$$ is $$2x+3y=9$$
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