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
$x \sin q \cdot \sin ^{2} q +y \cos ^{3} q=\sin q \cos q$
$\Rightarrow y \cos q \sin ^{2} q+ y \cos ^{3} q=\sin q \cos q$
$\Rightarrow y \cos q\left(\sin ^{2} q+\cos ^{2} q\right)=\sin q \cos q$
$\Rightarrow y \cos q=\sin q \cos q y=\sin q$ ... (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
$x^{2}+y^{2}=\cos ^{2} q+\sin ^{2}$
$q=1$