Trigonometric Equations Question 290
Question: If a, b and c are the sides of a triangle such that $ a^{4}+b^{4}+c^{4}=2c^{2}(a^{2}+b^{2}) $ then the angles opposite to the side C is
[J & K 2005]
Options:
A) $ 45{}^\circ $ or $ 135{}^\circ $
B) $ 30{}^\circ $ or $ 100{}^\circ $
C) $ 50{}^\circ $ or $ 100{}^\circ $
D) $ 60{}^\circ $ or $ 120{}^\circ $
Show Answer
Answer:
Correct Answer: A
Solution:
- $ a^{4}+b^{4}+c^{4}-2a^{2}c^{2}-2b^{2}c^{2}+2a^{2}b^{2}=2a^{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},or,\cos 135^{o}\Rightarrow C=45^{o}or13{5^{o}} $ .
BETA
