Question

Let $A(2,-3)$ and $B(-2,1)$ be vertices of a $△ABC$. If the centroid of $△ABC$ moves on the line $2x+3y=1$,then the locus of the vertex $C$ is

• A. $2x+3y=9$
• B. $2x-3y=7$
• C. $3x+2y=5$
• D. $3x-2y=3$

Answer 1

The correct option is A $2x+3y=9$

Let the vertex C be $(h,k)$

Given $A(2,-3),B(-2,1)$ and the centroid lies on $2x+3y=1$

$\therefore$ Let centroid be $(x,\frac{1-2k}{3})$

$\therefore x=\frac{2-2+h}{3};\frac{1-2k}{3}=\left(\frac{-2+k}{3}\right)$

$\therefore h=3x;3=2x+k$

$\therefore h=3x;k=3-2k$

$\therefore \frac{h}{3}=x$

$\therefore k=3-2\left(\frac{h}{3}\right)$

$\therefore 3k=9-2h$

$\therefore 2h+3k=9$

$\therefore 2x+3y=9$ is the locus.