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If $x^{2} + xy + xz = 135, y^{2} + yz + xy = 351, z^{2} + xz + yz = 243$ then what is the value of $x^{2} + y^{2} + z^{2}$ ?

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300

275

250

225

answer is B.

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Detailed Solution

{(x + y + z)}^{2} = x^{2} + y^{2} + z^{2} + 2 (xy + yz + zx)

x^{2} + xy + xz = 135, y^{2} + yz + xy = 351, z^{2} + xz + yz = 243

Adding all equations

x^{2} + y^{2} + z^{2} + 2 (xy + yz + zx) = 135 + 351 + 243 = 729 = 27^{2}

{(x + y + z)}^{2} = 27^{2}

(x + y + z) = 27

x^{2} + xy + xz = 135

\Rightarrow x (x + y + z) = 135

\Rightarrow x = \frac{135}{27} = 5

Similarly, y = 13 and z = 9.

x^{2} + y^{2} + z^{2} = 5^{2} + 13^{2} + 9^{2} = 25 + 169 + 81 = 275

So, Option 2 is correct.

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