Trigonometric Equations Question 179
Question: In a $ \Delta ABC, $ if $ \frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)}, $ then $ a^{2},\ b^{2},\ c^{2} $ are in
[Pb. CET 2001; Karnataka CET 1999]
Options:
A) A.P.
B) G.P.
C) H.P.
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
- $ \frac{\sin A}{\sin C}=\frac{\sin A\cos B-\cos A\sin B}{\sin B\cos C-\cos B\sin C} $
Þ $ \frac{a}{c}=\frac{a\cos B-b\cos A}{b\cos C-c\cos B} $ , (Using sine formula) Þ $ ,ab\cos C-ac\cos B=ac\cos B-bc\cos A $
Þ $ ab\cos C+bc\cos A=2ac\cos B $
Þ $ \frac{a^{2}+b^{2}-c^{2}}{2}+\frac{b^{2}+c^{2}-a^{2}}{2}=\frac{c^{2}+a^{2}-b^{2}}{1} $
Þ $ b^{2}=c^{2}+a^{2}-b^{2} $ Þ $ b^{2}=\frac{c^{2}+a^{2}}{2} $
$ \Rightarrow a^{2},,b^{2},,c^{2} $ are in A.P.
BETA
