If \( x\sin(a+y)+(\sin a)\cos(a+y)=0 \), then \( \displaystyle\frac{dy}{dx} \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{\sin^{2}(a+y)}{\sin a} \)
The relation can be solved for \( x \) directly:
\( \displaystyle x=-\sin a\cdot\frac{\cos(a+y)}{\sin(a+y)} \)
\( \displaystyle =-\sin a\cot(a+y) \)
Differentiating with respect to \( y \):
\( \displaystyle \frac{dx}{dy}=\sin a\,\mathrm{cosec}^{2}(a+y) \)
\( \displaystyle =\frac{\sin a}{\sin^{2}(a+y)} \)
Taking the reciprocal:
\( \displaystyle \frac{dy}{dx}=\frac{\sin^{2}(a+y)}{\sin a} \)
NCERT Exemplar:
Continuity and Differentiability
