Question

If a circle of constant radius $3k$ passes through the origin and meets the axes at $A$ and $B$, the locus of the centroid of $△OAB$ is

• A. $x^2+y^2=k^2$
• B. $x^2+y^2=2k^2$
• C. $x^2+y^2=3k^2$
• D. None of these

Answer 1

The correct option is D None of these

Let the coordinates of $A$ and $B$ are $(a,0)$ and $(0,b)$ respectively

Clearly, $△OAB$ is a right-angled triangle, the hypotenuse $AB$ is a diameter of the circle.

Thus $AB=2\left(3k\right)=6k$

In $△AOB$

$\Rightarrow {OA}^2+{OB}^2={AB}^2$

$\Rightarrow a^2+b^2=36k^2$ .....(i)

Let $(\alpha ,\beta )$ be the coodinates of the centriod of

$\Rightarrow △OAB$. Then, $\alpha =\frac{a}{3},\beta =\frac{b}{3}$
$\Rightarrow a=3\alpha ,b=3\beta$

Substituting the values of $a,b$ in Eq.(i) we get

$\Rightarrow 9{\alpha }^2+9{\beta }^2=36k^2$

$\Rightarrow {\alpha }^2+{\beta }^2=4k^2$

So, locus of $(\alpha ,\beta )$ is $x^2+y^2=4k^2$.