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Q. 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

J & K CETJ & K CET 2007Straight Lines

Solution:

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) $