A) \[{{45}^{o}}\] or \[{{90}^{o}}\]
B) \[{{30}^{o}}\] or \[{{135}^{o}}\]
C) \[{{45}^{o}}\] or \[{{135}^{o}}\]
D) \[{{60}^{o}}\] or \[{{120}^{o}}\]
Correct Answer: C
Solution :
Given, \[{{a}^{4}}+{{b}^{4}}+{{c}^{4}}=2{{c}^{2}}({{a}^{2}}+{{b}^{2}})\] ?..(i) \[\because \] \[\cos \,C=\frac{{{a}^{2}}+{{b}^{2}}-{{c}^{2}}}{2ab}\] \[\Rightarrow \]\[{{\cos }^{2}}C=\left[ \frac{{{a}^{4}}+{{b}^{4}}+{{c}^{4}}+2{{a}^{2}}{{b}^{2}}-2{{c}^{2}}({{a}^{2}}+{{b}^{2}})}{4{{a}^{2}}{{b}^{2}}} \right]\] \[=\left[ \frac{2{{c}^{2}}({{a}^{2}}+{{b}^{2}})+2{{a}^{2}}{{b}^{2}}-2{{c}^{2}}({{a}^{2}}+{{b}^{2}})}{4{{a}^{2}}{{b}^{2}}} \right]\] [from Eq. (i)] \[\Rightarrow \] \[{{\cos }^{2}}C=\frac{1}{2}\] \[\Rightarrow \] \[\cos \,C=\pm \frac{1}{\sqrt{2}}\] \[\Rightarrow \] \[\angle C={{45}^{o}}\] or \[{{135}^{o}}\]
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