Question

If $ x{{\sin }^{3}}\theta +y{{\cos }^{3}}\theta =\sin \theta \cos \theta $ and $ x\sin \theta =y\cos \theta , $ then $ {{x}^{2}}+{{y}^{2}} $ is

• A. 2
• B. 0
• C. 3
• D. 4
• E. 1

Answer 1

Answer: (E)

Given that, $ x{{\sin }^{3}}\theta +y{{\cos }^{3}}\theta =\sin \theta \cos \theta $ ..(i) and $ x\sin \theta =y\cos \theta $ ..(ii) On using Eqs. (i) and (ii), we get $ y\text{ }\cos \theta \text{ }{{\sin }^{2}}\theta +y\text{ }{{\cos }^{3}}\theta =\sin \theta \text{ }\cos \theta $
$ \Rightarrow $ $ y\text{ }cos\theta (si{{n}^{2}}\theta +co{{s}^{2}}\theta )=sin\theta \text{ }cos\theta $
$ \Rightarrow $ $ y=sin\text{ }\theta $
On putting the value of y in Eq (ii), we get $ x\text{ }sin\theta =sin\theta \text{ }cos\theta $
$ \Rightarrow $ $ x=cos\theta $