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  4. If x sin θ =y sin ( θ +(2π /3) )=z sin ( θ +(4π /3) ) then

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Q. If $ x\sin \theta =y\sin \left( \theta +\frac{2\pi }{3} \right)=z\sin \left( \theta +\frac{4\pi }{3} \right) $ then

Jharkhand CECEJharkhand CECE 2015

Solution:

We have, $ x\sin \theta =y\sin \left( \theta +\frac{2\pi }{3} \right)=z\sin \left( \theta +\frac{4\pi }{3} \right) $
$ \Rightarrow $ $ \frac{\sin \theta }{1/x}=\frac{\sin \left( \theta +\frac{2\pi }{3} \right)}{1/y}=\frac{\sin \left( \theta +\frac{4\pi }{3} \right)}{1/z} $
$ \Rightarrow $ $ \frac{\sin \theta }{1/x}=\frac{\sin \left( \theta +\frac{2\pi }{3} \right)}{1/y}=\frac{\sin \left( \theta +\frac{4\pi }{3} \right)}{1/z} $
$ =\frac{\sin \theta +\sin (\theta +2\pi /3)+\sin (\theta +4\pi /3)}{1/x+1/y+1/z} $
$ \Rightarrow $ $ \frac{\sin \theta }{1/x}=\frac{\sin (\theta +2\pi /3)}{1/y}=\frac{\sin (\theta +4\pi /3)}{1/z} $
$ =\frac{\sin \theta +2\sin (\pi +\theta )\cos \pi /3}{1/x+1/y+1/z} $
$ \Rightarrow $ $ \frac{\sin \theta }{1/x}\times \left( \frac{1}{x}+\frac{1}{y}+\frac{1}{z} \right)=0 $
$ \Rightarrow $ $ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0 $
$ \Rightarrow $ $ xy+yz+zx=0 $