Question

If the parametric equation of a curve is given by $x=\cos \theta+\log \tan \frac{\theta}{2}$ and $y=\sin \theta$, then the points for which $\frac{d y}{d x}=0$ are given by

• A. $\theta=\frac{ n \pi}{2}, n \in z$
• B. $\theta=(2 n+1) \frac{\pi}{2}, n \in z$
• C. $\theta=(2 n+1) \pi, n \in z$
• D. $\theta= n \pi, n \in z$

Answer 1

Answer: (D)

$\frac{ d x }{ d 0}=-\sin \theta+\frac{1}{\tan \left(\frac{\theta}{2}\right)} \cdot \sec ^{2}\left(\frac{\theta}{2}\right) \frac{1}{2}$
$=-\sin \theta+\frac{1}{2 \sin \left(\frac{\theta}{2}\right) \cos \left(\frac{\theta}{2}\right)}$
$=-\sin \theta+\frac{1}{\sin \theta}$
$=\frac{1-\sin ^{2} \theta}{\sin \theta} ; \frac{ dx }{ d \theta}=\frac{\cos ^{2} \theta}{\sin \theta} ; \frac{ dy }{ d \theta}=\cos \theta$
$\frac{ dy }{ dx }=0 ; \tan \theta=0$
$\theta= n \pi, n \in z$