Question

A rod of length $3l$ units, slides with its ends $A$ and $B$ on the $x$ and $y$ axes respectively. Then the locus of the centroid of $△OAB$ is
(Here, $O$ is the origin.)

• A. $x^2+y^2=l^2$
• B. $x^2+y^2=9l^2$
• C. $3x^2+3y^2=l^2$
• D. $x^2+y^2=3l^2$

Answer 1

The correct option is A $x^2+y^2=l^2$

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

Let $G(h,k)$ be the centroid of $△OAB$.
$\therefore h=\frac{a+0+0}{3}=\frac{a}{3},k=\frac{0+b+0}{3}=\frac{b}{3}$
$\Rightarrow a=3h,b=3k$

In $△OAB$,
$O{A}^2+O{B}^2=A{B}^2$
$\Rightarrow a^2+b^2=(3l{)}^2$
$\Rightarrow 9h^2+9k^2=9l^2$
$\Rightarrow h^2+k^2=l^2$
$\therefore$ Locus of $G(h,k)$ is $x^2+y^2=l^2$