A) \[45{}^\circ \] or \[135{}^\circ \]
B) \[30{}^\circ \] or \[100{}^\circ \]
C) \[50{}^\circ \] or \[100{}^\circ \]
D) \[60{}^\circ \] or \[120{}^\circ \]
Correct Answer: A
Solution :
\[{{a}^{4}}+{{b}^{4}}+{{c}^{4}}-2{{a}^{2}}{{c}^{2}}-2{{b}^{2}}{{c}^{2}}+2{{a}^{2}}{{b}^{2}}=2{{a}^{2}}{{b}^{2}}\] Þ \[{{({{a}^{2}}+{{b}^{2}}-{{c}^{2}})}^{2}}={{(\sqrt{2}ab)}^{2}}\Rightarrow {{a}^{2}}+{{b}^{2}}-{{c}^{2}}=\pm \sqrt{2}ab\] Þ \[\frac{{{a}^{2}}+{{b}^{2}}-{{c}^{2}}}{2ab}=\pm \frac{\sqrt{2}ab}{2ab}=\pm \frac{1}{\sqrt{2}}\] Þ \[\cos C=\cos {{45}^{o}}\,\text{or}\,\cos {{135}^{o}}\Rightarrow C={{45}^{o}}\,\,\text{or}\,\,\text{13}{{\text{5}}^{o}}\].
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