Question

Let A (2,-3) and B(-2,1) be the vertices of a tirangle 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. 5x-3y=7
• C. 3x+2y=5
• D. 3x-2y=3

Answer 1

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

Let (x,y) be the coordinates of the vertex C and $(x_1,y_1)$ be the coordinates of the centroid of the triangle, therefore
$x_1=\frac{x+2-2}{3}{y}_1=\frac{y-3+1}{3}$
or $x_1=\frac{x}{3}$
and $y_1=\frac{y-2}{3}$
the centroid lies on the line $2x+3y=1.So{x}_{1}and{y}_1$ satisfy the equation of the line. Thus
$2x_1+3y_1=1\Rightarrow 2\left(\frac{x}{3}\right)+2\left(\frac{y-2}{3}\right)=12x+3y=9$
The above equation is the locus of the vertex C.