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Question

If $$\displaystyle x = y \cos {\dfrac {2\pi}{3}} = z \cos {\dfrac {4\pi}{3}}$$, then $$xy + yz + zx =$$


A
1
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B
0
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C
1
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D
2
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Solution

The correct option is B $$0$$
We have $$x = y \cos \dfrac {2\pi}{3} = z \cos \dfrac {4\pi}{3}$$ (say)

$$\Rightarrow \dfrac {1}{x} = \dfrac {1}{k}, \dfrac {1}{y} = \dfrac {\cos \dfrac {2\pi}{3}}{h}, \dfrac {1}{z} = \dfrac {\cos \dfrac {4\pi}{3}}{k}$$

$$\therefore \dfrac {1}{x} + \dfrac {1}{y} + \dfrac {1}{z} = \dfrac {1}{k}\left (1 + \cos \dfrac {2\pi}{3} + \cos \dfrac {4\pi}{3}\right ) = \dfrac {1}{k} \left (1 - \dfrac {1}{2} - \dfrac {1}{2}\right ) = 0$$

$$\Rightarrow xy + yz + xz = 0$$

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