Complex Numbers And Quadratic Equations question 130
Question: If $ a=\cos \alpha +i\sin \alpha ,b=\cos \beta +i\sin \beta , $ $ c=\cos \gamma +i\sin \gamma and\frac{b}{c}+\frac{c}{a}+\frac{a}{b}=1, $ then $ \cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta ) $ is equal to [RPET 2001]
Options:
A) 3/2
B) - 3/2
C) 0
D) 1
Show Answer
Answer:
Correct Answer: D
Solution:
$ \frac{b}{c}=\frac{\cos \beta +i\sin \beta }{\cos \gamma +i\sin \gamma }\times \frac{\cos \gamma -i\sin \gamma }{\cos \gamma -i\sin \gamma } $
$ \Rightarrow \frac{b}{c}=\cos (\beta -\gamma )+i\sin (\beta -\gamma ) $ ……(i) Similarly, $ \frac{c}{a}=\cos (\gamma -\alpha )+i\sin (\gamma -\alpha ) $ ……(ii) and $ \frac{a}{b}=\cos (\alpha -\beta )+i\sin (\alpha -\beta ) $ …..(iii) from (i) + (ii) + (iii) $ \cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta ) $ $ +i[\sin (\beta -\gamma )+\sin (\gamma -\alpha )+\sin (\alpha -\beta )]=1 $ Equating real and imaginary parts, $ \cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )=1 $ .
BETA
