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If xcostheta=ycos(theta+(2pi)/3)=z cos(t...

If `xcostheta=ycos(theta+(2pi)/3)=z cos(theta+(4pi)/3)` , prove that `x y+y z+z x=0.`

Text Solution

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Let `x cos theta=y cos(theta+(2pi)/(3))=z cos(theta+(4pi)/(3))=k`
Now, `xy+yz+zx=xyz((1)/(x)+(1)/(y)+(1)/(z))`
`=(xyz)/(4)(cos theta+{cos(theta+(2pi)/(3))+cos(theta+(4pi)/(3))})`
`=(xyz)/(4k)(cos theta+2cos((pi)/(3))cos (theta+pi))`
`=(xyz)/(k)(cos theta-2(1)/(2)cos theta)=0`
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