Question

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

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

Answer 1

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

Here, $A\equiv (2,-3)$ and $B\equiv (-2,1)$
Let $C\equiv (h,k)$
So, the centroid of $\Delta ABC$ is $\equiv (\frac{2+(-2)+h}{3},\frac{-3+1+k}{3})\equiv G$ (say)
$\therefore G\equiv (\frac{h}{3},\frac{k-2}{3})$
Now given $G$ lies on the line $2x+3y=1$
$\Rightarrow 2\cdot \frac{h}{3}+3\cdot \frac{k-2}{3}=1$
$\Rightarrow 2h+3k=3+6=9$
Hence, required locus of $P(h,k)$ is $2x+3y=9$.