If $x + y + z = 0$ then $x^{2} + x y + y^{2}$ equals

y^{2} + y z + z^{2}

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y^{2} - y z + z^{2}

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z^{2} - z x + x^{2}

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z^{2} + z x + x^{2}

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Solution

The correct option is C $z^{2} + z x + x^{2}$
$x + y + z = 0$
$\Rightarrow x + y = - z$ and $y = - z - x$
Now, $x^{2} + x y + y^{2} = x^{2} + 2 x y + y^{2} - x y$
= ${(x + y)}^{2} - x y = {(- z)}^{2} - x (- z - x)$
= $z^{2} + x z + x^{2}$

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