If `asintheta="bsin"(theta+(2pi)/3)="csin"(theta+(4pi)/3)` , prove that `a b+b c+c a=0.`

Options

A. `x+y+z=0`
B. `xy+yz+zx=0`
C. `xyz+x+y+z=1`
D. none of these

Answer

B

Explanation

We have,
`x sintheta=y sin(theta+(2pi)/(3))=zsin(theta+(4pi)/(3))`
`implies(sin theta)/(1/x)=(sin (theta+(2pi)/(3)))/(1/y)=(sin(theta+(4pi)/(3)))/(1/z)`
`=(sintheta)/(1/x)=(sin(theta+(2pi)/(3)))/(1/y)=(sin(theta+(4pi)/(3)))/(1/z)`
`=(sin theta+sin(theta+(2pi)/(3))+sin(theta+(4pi)/(3)))/(1/x+1/y+1/z)`
`implies(sintheta)/(1/x)=(sin(theta+(2pi)/(3)))/(1/y)=(sin(theta+(4pi)/(3)))/(1/z)`
`implies(sintheta)/(1/x)xx((1)/(x)+(1)/(y)+(1)/(z))=0`
`implies1/x+1/y+1/z=0impliesxy+yz+zx=0`