Question

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

• A. $-4x^2$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is B $0$

$x=y\cos 2\pi /3=z\cos 4\pi /3$

$x=y\cos (\pi -\pi /3)$, $x=z\cos (\pi +\pi /3)$

$x=-y\cos \pi /3$, $x=-z\cos \pi /3$

$x=-y\times \frac{1}{2}$, $x=-z\times \frac{1}{2}$

$x=-y/2$, $x=-z/2$

$\Rightarrow -y/2=-z/2$ or $y=z$

$\therefore x=-y/2$, $y=y_1,z=y$

Now $xy+yz+zx=\frac{-y}{2}\cdot y+y\cdot y+y\cdot -y/2$

$=\frac{-y^2}{2}+y^2-\frac{y^2}{2}=-y^2+y^2$

$=0$.