Question

If the centroid of the triangle formed by the points $(0,0),(\cos \theta ,\sin \theta )$ and $(sin\theta ,-\cos \theta )$ lies on the line $y=2x,$ then $\theta$ is equal to

• A. ${\tan }^{-1}2$
• B. ${\tan }^{-1}3$
• C. ${\tan }^{-1}(-3)$
• D. ${\tan }^{-1}(-2)$

Answer 1

Answer: (C)

Given vertices of triangle are $O(0,0),$
$A(\cos \theta ,\sin \theta )$ and $B(sin\theta ,-cos\theta )$
coordinate of centroid are
$\left(\frac{\cos \theta +\sin \theta }{3},\frac{\sin \theta -\cos \theta }{3}\right)$ .
Since, centroid lies on the line
$y=2x.$
$\therefore$ $\frac{\sin \theta -\cos \theta }{3}=\frac{2\cos \theta +2\sin \theta }{3}$
$\Rightarrow$ $\sin \theta =-3\cos \theta$
$\Rightarrow$ $\tan \theta =-3$
$\Rightarrow$ $\theta ={\tan }^{-1}(-3)$