A) \[({{y}^{2}}+yx+{{z}^{2}})\]
B) \[({{y}^{2}}-yz+{{z}^{2}})\]
C) \[({{z}^{2}}-zx+{{x}^{2}})\]
D) \[({{z}^{2}}+zx+{{x}^{2}})\]
Correct Answer: D
Solution :
\[x+y+z=0\] \[\Rightarrow \,\] \[\left( x+y \right)=-z\Rightarrow {{\left( x+y \right)}^{2}}={{z}^{2}}\] \[\Rightarrow \,\]\[{{x}^{2}}+{{y}^{2}}+2xy={{z}^{2}}\] \[\Rightarrow \,\]\[{{x}^{2}}+{{y}^{2}}+xy={{z}^{2}}-xy\] \[\,[\therefore x+y+z=0\Rightarrow y=(-x-z)]\] \[={{z}^{2}}-x(-x-z)\] \[\Rightarrow \,\] \[{{x}^{2}}+{{y}^{2}}+xy={{z}^{2}}+zx+{{x}^{2}}\]
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