Question

Statement $I$: $A$($0,0$), $B(\cos \alpha ,\sin \alpha )$ ,$C(-\sin \alpha ,\cos \alpha )$ are vertices of a triangle then the locus of the centroid of triangle is $9x^2+9y^2=4$

Statement $II$: The locus of the point ($a\cos \theta$, $b\sin \theta$) is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Which of the above statement is/are correct:

• A. Only $I$
• B. Only $II$
• C. Both $I$ and $II$
• D. Neither $I$ Nor $II$

Answer 1

Answer: (B)

Only $II$

Let the centroid of the triangle be Q (h,k)

then using centroid formulaes we get the co-ordiantes as,

$h=\frac{0+cos\alpha -sin\alpha }{3}$

$3h=cos\alpha -sin\alpha$
$3k=cos\alpha +sin\alpha$
$\frac{3(h+k)}{2}=cos\alpha$
$\frac{3(k-h)}{2}=sin\alpha$
$\frac{3^2(k+h{)}^2}{2^2}+\frac{3^2(k-h{)}^2}{2^2}=1$
$9x^2+9y^2=4$
$\frac{(acos\theta {)}^2}{a^2}+\frac{(bsin\theta {)}^2}{b^2}=1$
$co{s}^2\theta +si{n}^2\theta =1$