Question

A straight rod of lengh $9units$ slides with it's end A and B on the coordinate axes. Then find the locus of the centroid of the $△OAB$, where O is the origin.

Answer 1

Let the coordinates of ends of rod be $A(a,0),B(0,b)$

$AB=9⟹a^2+b^2=81\cdots \left(1\right)$

Let the centroid of the $△OAB$ be $(h,k)$

$h=\frac{0+a+0}{3}=\frac{a}{3},k=\frac{0+0+b}{3}=\frac{b}{3}$

$⟹a=3h,b=3k$

From $\left(1\right),(3h{)}^2+(3k{)}^2=81⟹h^2+k^2=9$

The locus of centroid of $△OAB$ is $x^2+y^2=9$