Question

The equation $ a\text{ }cos \theta +b\text{ }sin \theta =c $ has a solution, when $ a $ , $ b $ and $ c $ are real numbers such that:

• A. $ a < b < c $
• B. $ a=b=c $
• C. $ {{c}^{2}}\le {{a}^{2}}+{{b}^{2}} $
• D. $ {{c}^{2}}<{{a}^{2}}-{{b}^{2}} $
• E. for all real values of a, b and c

Answer 1

Answer: (C)

The given equation is
$ a\text{ }cos\theta +b\text{ }sin\theta =c $
Since, $ \sqrt{{{a}^{2}}-{{b}^{2}}}\le a\cos \theta +b\sin \theta \le \sqrt{{{a}^{2}}+{{b}^{2}}} $
$ \Rightarrow $ $ {{c}^{2}}\le \sqrt{{{a}^{2}}+{{b}^{2}}} $