Question

Find locus of centroid of $\Delta AOB$ if line AB passes through $(3,2),A$ and $B$ are on the coordinate axes.

Answer 1

Given A and B are on the coordinate axes so

Let the coordinates of A be $(x,0)$ and B be $(0,y)$

We know centroid of the triangle is the sum of the coordinates divided by $3$

i.e. centroid =$(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$ where $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ are the coordinates of the

triangle therefore centroid of triangle AOB is $(\frac{0+x+0}{3},\frac{0+y+0}{3})$ i.e $(\frac{x}{3},\frac{y}{3})$ now given line AB passes through $(3,2)$

Now the equation of the line is $Y-0=\frac{y-0}{0-x}(X-x)$

$Y(-x)=y(X-x)$

in this equation if we substitute $(3,2)$ in place of $(X,Y)$ we get the equation

$\Rightarrow 2x+3y=xy$

substituting $3x,3y$ in place of $x,y$ we get the locus of the centroid

$\therefore$ The locus of the centroid is $2X+3Y=3XY$