A) \[x+y+z=0\]
B) \[xy+yz+zx=0\]
C) \[xyz+x+y+z=1\]
D) None of the above
Correct Answer: B
Solution :
We have, \[x\sin \theta =y\sin \left( \theta +\frac{2\pi }{3} \right)=z\sin \left( \theta +\frac{4\pi }{3} \right)\] \[\Rightarrow \]\[\frac{\sin \theta }{1/x}=\frac{\sin \left( \theta +\frac{2\pi }{3} \right)}{1/y}=\frac{\sin \left( \theta +\frac{4\pi }{3} \right)}{1/z}\] \[\Rightarrow \]\[\frac{\sin \theta }{1/x}=\frac{\sin \left( \theta +\frac{2\pi }{3} \right)}{1/y}=\frac{\sin \left( \theta +\frac{4\pi }{3} \right)}{1/z}\] \[=\frac{\sin \theta +\sin (\theta +2\pi /3)+\sin (\theta +4\pi /3)}{1/x+1/y+1/z}\] \[\Rightarrow \]\[\frac{\sin \theta }{1/x}=\frac{\sin (\theta +2\pi /3)}{1/y}=\frac{\sin (\theta +4\pi /3)}{1/z}\] \[=\frac{\sin \theta +2\sin (\pi +\theta )\cos \pi /3}{1/x+1/y+1/z}\] \[\Rightarrow \] \[\frac{\sin \theta }{1/x}\times \left( \frac{1}{x}+\frac{1}{y}+\frac{1}{z} \right)=0\] \[\Rightarrow \] \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\] \[\Rightarrow \] \[xy+yz+zx=0\]
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