Question

Assertion :If $x\cos \theta =y\cos ({120}^o+\theta )=z\cos (\theta +{240}^o)$, then $xy+yz+zx=1$ Reason: $\cos \alpha +\cos ({120}^o+\alpha )+\cos ({120}^o-\alpha )=0$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let

$x\cos \theta =y\cos (120+\theta )=z\cos (\theta +240)=\frac{1}{k}$

$\Rightarrow \frac{\frac{1}{x}}{\cos \theta }=\frac{\frac{1}{y}}{\cos (120+\theta )}=\frac{\frac{1}{z}}{\cos (\theta +240)}=k$

$\frac{\frac{xy+yz+zx}{xyz}}{\cos \theta +\cos (\theta +240)+\cos (120+\theta )}=k$

$\frac{\frac{xy+yz+zx}{xyz}}{2\cos 120+\cos (\theta +120)+\cos (120+\theta )}=k$

$\frac{xy+yz+zx}{xyz\{-\cos (\theta +120)+\cos (\theta +120\left)\right\}}=k$

$\frac{xy+yz+zx}{0}=k$

$xy+yz+zx=0$