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If a circle of a constant radius 6 passes through origin $O$ and meets the coordinate axes at $A$ and $B$, then find the locus of the centroid of $\triangle O A B$.
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The correct answer is:
$x^2+y^2=16$
Let $(h, k)$ be centroid of triangle
$\therefore \quad h=\frac{a}{3}$ and $k=\frac{b}{3}$
$a=3 h, b=3 k$
Now, $A B$ is diameter of circle.
$\begin{aligned} \therefore & a^2+b^2=144 \\ \therefore & (3 h)^2+(3 k)^2=144 \Rightarrow h^2+k^2=\frac{144}{9}\end{aligned}$
$h^2+k^2=16$
$\therefore$ Locus of centroid of $\triangle O A B$ is $x^2+y^2=16$
$\therefore \quad h=\frac{a}{3}$ and $k=\frac{b}{3}$
$a=3 h, b=3 k$
Now, $A B$ is diameter of circle.
$\begin{aligned} \therefore & a^2+b^2=144 \\ \therefore & (3 h)^2+(3 k)^2=144 \Rightarrow h^2+k^2=\frac{144}{9}\end{aligned}$
$h^2+k^2=16$
$\therefore$ Locus of centroid of $\triangle O A B$ is $x^2+y^2=16$
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