Question

State the whether given statement is true or false

$x\cos \theta =y\cos (\theta +\frac{2\pi }{3})=z\cos (\theta +\frac{4\pi }{3})$,then $xy+yz+zx=0$.

• A. True
• B. False

Answer 1

The correct option is A True

Let $x\cos \theta =y\cos (\theta +\frac{2\pi }{3})=z\cos (\theta +\frac{4\pi }{3})=k$ ....... (1)

Dividing $xy+yz+xz$ by $xyz$ we get

$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$

$=\frac{1}{k}[\cos \theta +\cos (\theta +\frac{2\pi }{3})+\cos (\theta +\frac{4\pi }{3})]$ ...... from(1)

$=\frac{1}{k}\left[\cos \theta +2\cos (\pi +\theta )\cos \frac{\pi }{3}\right]$

$=\frac{1}{k}[\cos \theta -\cos \theta ]=0$