If $x, y, z$ are in A.P.,then the value of $(x + y - z) (y + z - x)$ is equal to

8 y z - 3 y^{2} - 4 z^{2}

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8 y z - 3 z^{2} - 4 y^{2}

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8 y z + 3 y^{2} - 4 z^{2}

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8 y z - 3 y^{2} + 4 z^{2}

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Solution

The correct option is A $8 y z - 3 y^{2} - 4 z^{2}$
If $x, y$ and $z$ are in AP, then $2 y = x + z$
Substituting $x$ as $2 y - z$ in the above expression, we get
$(x + y - z) (y + z - x)$
$= (2 y - z + y - z) (y + z - 2 y + z)$
$= (3 y - 2 z) (- y + 2 z)$
$= 6 y z - 3 y^{2} - 4 z^{2} + 2 y z$
$= 8 y z - 3 y^{2} - 4 z^{2}$

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