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Question Number 5935 by Master Tria last updated on 05/Jun/16

If a, b, c, x, y, z ∈ R be such that (a+b+c)^2 =3(ab+bc+ca−x^2 −y^2 −z^2 ), then

$$\mathrm{If}\:{a},\:{b},\:{c},\:{x},\:{y},\:{z}\:\in\:{R}\:\mathrm{be}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\left({a}+{b}+{c}\right)^{\mathrm{2}} =\mathrm{3}\left({ab}+{bc}+{ca}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} −{z}^{\mathrm{2}} \right), \\ $$$$\mathrm{then} \\ $$

Commented by prakash jain last updated on 07/Jun/16

Is it an objective question with opions?

$$\mathrm{Is}\:\mathrm{it}\:\mathrm{an}\:\mathrm{objective}\:\mathrm{question}\:\mathrm{with}\:\mathrm{opions}? \\ $$

Answered by Yozzii last updated on 05/Jun/16

(a+b+c)^2 =3(ab+bc+ca−x^2 −y^2 −z^2 ) (a+b+c)^2 −2(ab+bc+ca)=ab+bc+ca−3(x^2 +y^2 +z^2 ) a^2 +b^2 +c^2 =ab+bc+ca−3(x^2 +y^2 +z^2 ) a^2 +b^2 +c^2 −ab−bc−ca=−3(x^2 +y^2 +z^2 ) (a+b+c)(a^2 +b^2 +c^2 −ab−bc−ca)+3abc=3abc−3(a+b+c)(x^2 +y^2 +z^2 ) a^3 +b^3 +c^3 =3(abc−(a+b+c)(x^2 +y^2 +z^2 ))

$$\left({a}+{b}+{c}\right)^{\mathrm{2}} =\mathrm{3}\left({ab}+{bc}+{ca}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} −{z}^{\mathrm{2}} \right) \\ $$$$\left({a}+{b}+{c}\right)^{\mathrm{2}} −\mathrm{2}\left({ab}+{bc}+{ca}\right)={ab}+{bc}+{ca}−\mathrm{3}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right) \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} ={ab}+{bc}+{ca}−\mathrm{3}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right) \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}=−\mathrm{3}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right) \\ $$$$\left({a}+{b}+{c}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}\right)+\mathrm{3}{abc}=\mathrm{3}{abc}−\mathrm{3}\left({a}+{b}+{c}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right) \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\mathrm{3}\left({abc}−\left({a}+{b}+{c}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)\right) \\ $$$$ \\ $$

Commented by Yozzii last updated on 05/Jun/16

I′m unsure of what is being asked of the question.

$${I}'{m}\:{unsure}\:{of}\:{what}\:{is}\:{being}\:{asked} \\ $$$${of}\:{the}\:{question}. \\ $$

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