A) \[\sqrt{2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}\]
B) \[\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\]
C) \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
D) \[{{({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}^{2}}\]
Correct Answer: A
Solution :
\[\frac{{{x}^{2}}-2{{a}^{2}}}{{{b}^{2}}+{{c}^{2}}}+\frac{{{x}^{2}}-2{{b}^{2}}}{{{c}^{2}}+{{a}^{2}}}=0+\frac{{{x}^{2}}-2{{c}^{2}}}{{{a}^{2}}+{{b}^{2}}}=6\] \[\Rightarrow \]\[\frac{{{x}^{2}}-2{{a}^{2}}}{{{b}^{2}}+{{c}^{2}}}-2+\frac{{{x}^{2}}-2{{b}^{2}}}{{{c}^{2}}+{{a}^{2}}}-2+\frac{{{x}^{2}}-2{{c}^{2}}}{{{a}^{2}}+{{b}^{2}}}-2=0\] \[\Rightarrow \]\[\frac{{{x}^{2}}-2{{a}^{2}}-2{{b}^{2}}-2{{c}^{2}}}{{{b}^{2}}+{{c}^{2}}}+\frac{{{x}^{2}}-2{{b}^{2}}-2{{c}^{2}}-2{{a}^{2}}}{{{c}^{2}}+{{a}^{2}}}\]\[+\frac{{{x}^{2}}-2{{c}^{2}}-2{{a}^{2}}-2{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}=0\] \[\Rightarrow \]\[({{x}^{2}}-2{{a}^{2}}-2{{b}^{2}}-2{{c}^{2}})\] \[\left[ \frac{1}{{{b}^{2}}+{{c}^{2}}}+\frac{1}{{{c}^{2}}+{{a}^{2}}}+\frac{1}{{{a}^{2}}+{{b}^{2}}} \right]=0\] \[\Rightarrow \] \[{{x}^{2}}-2{{a}^{2}}-2{{b}^{2}}-2{{c}^{2}}=0\] \[\Rightarrow \] \[{{x}^{2}}=2{{a}^{2}}+2{{b}^{2}}+2{{c}^{2}}\] \[\Rightarrow \] \[x=\pm \sqrt{2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}\]
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