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Question

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


Solution

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 =$$\left (\dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3}\right )$$ 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 $$(\dfrac{0+x+0}{3},\dfrac{0+y+0}{3})$$ i.e $$(\dfrac x3,\dfrac y3)$$ now given line AB passes through $$(3,2)$$

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

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 $$2X+3Y=3XY$$

Mathematics

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