Question

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

• A. 1
• B. 2
• C. 0
• D. None of these

Answer 1

Answer: (A)

$x{\sin }^{3}q+y{\cos }^{3}q=\sin q\cos q$ ... (i)
and $x\sin q=y\cos q$ ... (ii)
Equation (i) may be written as
$x\sin q\cdot {\sin }^{2}q+y{\cos }^{3}q=\sin q\cos q$
$\Rightarrow y\cos q{\sin }^{2}q+y{\cos }^{3}q=\sin q\cos q$
$\Rightarrow y\cos q\left({\sin }^{2}q+{\cos }^{2}q\right)=\sin q\cos q$
$\Rightarrow y\cos q=\sin q\cos qy=\sin q$ ... (iii)
Putting the value of $y$ from (iii) in (ii),
we get $x\sin q=\sin q\cdot \cos q$
$\Rightarrow x=\cos q$ ... (iv)
Squaring (iii) and (iv) and adding,
we get $x^2+y^2={\cos }^{2}q+{\sin }^{2}q=1$
$x\sin q=\sin q.\cos q$
$\Rightarrow x=\cos q$ ... (iv)
Squaring (iii) and (iv) and adding, we get
$x^2+y^2={\cos }^{2}q+{\sin }^2$
$q=1$