A) \[2\text{ }{{a}^{2}}({{b}^{2}}+{{c}^{2}})\]
B) \[2{{c}^{2}}({{a}^{2}}+{{b}^{2}})\]
C) \[2{{b}^{2}}({{a}^{2}}+{{c}^{2}})\]
D) \[2({{a}^{2}}{{b}^{2}}+{{b}^{2}}{{c}^{2}}+{{c}^{2}}{{a}^{2}})\]
E) \[2{{a}^{2}}{{b}^{2}}+2{{b}^{2}}{{c}^{2}}+3{{a}^{2}}{{c}^{2}}\]
Correct Answer: C
Solution :
\[\cos B=\frac{{{c}^{2}}+{{a}^{2}}-{{b}^{2}}}{2ac}\] \[\Rightarrow \] \[\frac{1}{\sqrt{2}}=\frac{{{c}^{2}}+{{a}^{2}}-{{b}^{2}}}{2ac}\] \[\Rightarrow \] \[\sqrt{2}ac={{c}^{2}}+{{a}^{2}}-{{b}^{2}}\] On squaring both sides, we get \[2{{a}^{2}}{{c}^{2}}={{c}^{4}}+{{a}^{4}}+{{b}^{4}}+2{{c}^{2}}{{a}^{2}}-2({{c}^{2}}+{{a}^{2}}){{b}^{2}}\] \[\Rightarrow \] \[{{c}^{4}}+{{a}^{4}}+{{b}^{2}}=2({{c}^{2}}+{{a}^{2}}){{b}^{2}}\]
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