Question

Consider the circle $x^2+y^2=a^2$. Let $A(a,0)$ and $D$ be a given interior point of the circle. If $BC$ be an arbitrary chord of the circle through point $D$, then the locus of the centroid of $\Delta ABC$ is

• A. a circle whose radius is less than $\frac{2a}{3}$ units
• B. a circle whose radius is greater than $\frac{2a}{3}$ units
• C. a circle whose radius is equal to $\frac{2a}{3}$ units
• D. None of the above

Answer 1

The correct option is A a circle whose radius is less than $\frac{2a}{3}$ units

$\because$ $B,C$ lies on the circle
$\therefore$ parametric coordinate of
$B\equiv (a\cos \alpha ,a\sin \alpha ),C\equiv (a\cos \beta ,a\sin \beta )$
where $\alpha ,\beta ,(\alpha \ne \beta )$ are parametric angles
and $A\equiv (a,0)$
Let the coordinates of centriod of $△ABC$ be $G(h,k)$
$\therefore h=\frac{a+a\cos \alpha +a\cos \beta }{3},k=\frac{a\sin \alpha +a\sin \beta }{3}$
$\Rightarrow {(h-\frac{a}{3})}^2+k^2=\frac{a^2}{9}(1+1+2\cos (\alpha -\beta \left)\right)$
Hence locus will be
${(x-\frac{a}{3})}^2+y^2=\frac{a^2}{9}(2+2\cos (\alpha -\beta \left)\right)$
Hence locus will be circle and radius will be
$<2a/3(\because 2+2\cos (\alpha -\beta )<4)$