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

Solution:

We have,

$\begin{array}{l}x=y\cos \frac{2\pi }{3}=z\cos \frac{4\pi }{3}\\ \Rightarrow x=-\frac{y}{2}=-\frac{z}{2}\\ \Rightarrow \frac{x}{1}=\frac{y}{-2}=\frac{z}{-2}=\lambda \text{ (say) }\\ \Rightarrow x=\lambda ,y=-2\lambda ,z=-2\lambda \\ \Rightarrow xy+yz+zx=-2{\lambda }^2+4{\lambda }^2-2{\lambda }^2=0\end{array}$