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 $y\cos \theta {\sin }^2\theta +y{\cos }^3\theta =\sin \theta \cos \theta$
$\Rightarrow$ $ycos\theta (si{n}^2\theta +co{s}^2\theta )=sin\theta cos\theta$
$\Rightarrow$ $y=sin\theta$
On putting the value of y in Eq (ii), we get $xsin\theta =sin\theta cos\theta$
$\Rightarrow$ $x=cos\theta$