Question

lf $A=(aCos\theta ,bSin\theta )$ ,$B=(-aSin\theta ,bCos\theta )$ , $O$ is the origin and $\theta$ is a parameter, then the locus of the centroid of $\Delta A$O$B$ is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=$

• A. 2/9
• B. 1/9
• C. 9/2
• D. 1

Answer 1

The correct option is A 2/9

Let P(h,k) be the centroid then,

using the formula for the centroid of the traingle we get,

$h=\frac{acos\theta -asin\theta +0}{3}$
$k=\frac{bsin\theta +bcos\theta +0}{3}$
$3\frac{h}{a}=cos\theta -sin\theta$
$3\frac{k}{b}=sin\theta +cos\theta$

now squaring and adding the above two equations,
$\frac{9h^2}{a^2}+\frac{9k^2}{b^2}=2$
$\frac{h^2}{a^2}+\frac{k^2}{b^2}=\frac{2}{9}$
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{2}{9}$