Question
If the points $$\left( { - 2,\,0} \right),\,\left( { - 1,\,\frac{1}{{\sqrt 3 }}} \right)$$ and $$\left( {\cos \,\theta ,\,\sin \,\theta } \right)$$ are collinear then the number of values of $$\theta \, \in \left[ {0,\,2\pi } \right]$$ is :
A.
0
B.
1
C.
2
D.
infinite
Answer :
1
Solution :
\[\begin{array}{l}
\frac{1}{2}\left| \begin{array}{l}
\,\, - 2\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\
\,\, - 1\,\,\,\,\,\,\,\,\,\frac{1}{{\sqrt 3 }}\,\,\,\,\,\,\,\,\,\,\,1\\
\cos \,\theta \,\,\,\,\sin \,\theta \,\,\,\,\,\,\,1
\end{array} \right| = 0\\
\Rightarrow \sqrt 3 \sin \,\theta - \cos \,\theta = 2\\
\Rightarrow \sin \left( {\theta - \frac{\pi }{6}} \right) = 1\\
\Rightarrow \theta - \frac{\pi }{6} = \frac{\pi }{2}
\end{array}\]