If $x - y + z = 5$ and $x^{2} + y^{2} + z^{2} = 49$ , then the value of $z x - x y - y z$ is

12

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- 12

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37

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- 37

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Solution

The correct option is D $- 12$
Given,
$x - y + z = 5$ and $x^{2} + y^{2} + z^{2} = 49$
We know,
$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)$
Here,
$a = x$
$b = - y$
$c = z$
Thus,
$\Rightarrow (x - y + z)^{2} = x^{2} + y^{2} + z^{2} + 2 (- x y - y z + z x)$
$\Rightarrow 5^{2} = 49 + 2 (- x y - y z + z x)$
$\Rightarrow - 2 (- x y - y z + z x) = 49 - 25$
$\Rightarrow (- x y - y z + z x) = - \frac{24}{2}$
$\Rightarrow (- x y - y z + z x) = - 12$

Hence, option B.

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