Question

The centroid of a triangle formed by the points $(0,0),(\cos \theta ,\sin \theta )$ and $(\sin \theta ,-\cos \theta )$ lie on the line $y=2x$; then $\theta$ is.

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

Answer 1

The correct option is C

${\tan }^{-1}-3$

Explanation for the correct option

Step 1: Find the centroid

Given vertices of a triangle are $O(0,0),A(\cos \theta ,\sin \theta )$ and $B(\sin \theta ,-\cos \theta )$.

Draw the diagram

If $P$ is the centroid of a $△OAB$, then $P$ divides the median in $2:1$.

Let $D$ is the mid-point of $\mathrm{AB}$.
$\mathrm{D}=\left[\frac{\sin \theta +\cos \theta }{2},\frac{\sin \theta -\cos \theta }{2}\right]$
If $\mathrm{P}$ is centroid of a $∆\mathrm{OAB}$, then the coordinates of $\mathrm{P}$ is:
$$\begin{gathered} P=\left[0+\frac{2}{3}\left(\frac{\sin \theta +\cos \theta }{2}\right),0+\frac{2}{3}\left(\frac{\sin \theta -\cos \theta }{2}\right)\right] \\ P=\left(\frac{\sin \theta +\cos \theta }{3},\frac{\sin \theta -\cos \theta }{3}\right) \end{gathered}$$]

Step 2: Find the value of $\theta$
Also given centroid lies on the line $y=2x$.
$$\begin{gathered} \Rightarrow \frac{\sin \theta -\cos \theta }{3}=2\left[\frac{\sin \theta +\cos \theta }{3}\right] \\ \Rightarrow \sin \theta -\cos \theta =2\sin \theta +2\cos \theta \\ \Rightarrow -\sin \theta =3\cos \theta \\ \Rightarrow \tan \theta =-3 \\ \Rightarrow \theta ={\tan }^{-1}(-3) \end{gathered}$$
Hence, option (C) is correct.