Limits, Continuity and Differentiability
80294
If \(x=a(1-\cos \theta), y=a(\theta-\sin \theta)\), then \(\frac{d^{2} y}{d x^{2}}=\)
1 \(\frac{\sin \left(\frac{\theta}{2}\right)}{2 \mathrm{a} \cos \theta}\)
2 \(\frac{\cos ^{2}\left(\frac{\theta}{2}\right)}{2 \mathrm{a} \operatorname{cosec} \theta}\)
3 \(\frac{\operatorname{cosec} \theta}{2 \mathrm{a} \cos ^{2}\left(\frac{\theta}{2}\right)}\)
4 \(\frac{\cos \left(\frac{\theta}{2}\right)}{2 \mathrm{a} \sin \theta}\)
Explanation:
(C) : Given
\(\mathrm{x}=\mathrm{a}(1-\cos \theta)\)
On differentiating both sides w.r.t. \(\theta\)
\(\frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{a}(0+\sin \theta)\)
\(\frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{a} \sin \theta\)
And, \(\mathrm{y}=\mathrm{a}(\theta-\sin \theta)\)
\(\frac{d y}{d \theta}=a(1-\cos \theta)\)
\(\therefore \quad \frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}\)
\(\frac{d y}{d x}=\frac{a(1-\cos \theta)}{a \sin \theta} \Rightarrow \frac{d y}{d x}=\frac{1-\cos \theta}{\sin \theta}\)
\(\frac{d y}{d x}=\frac{1-1+2 \sin ^{2} \frac{\theta}{2}}{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}} \Rightarrow \frac{d y}{d x}=\tan \frac{\theta}{2}\)
Again, differentiating both sides w.r. t. \(\mathrm{x}\)
\(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\mathrm{d}}{\mathrm{d} \theta}\left(\tan \frac{\theta}{2}\right) \times \frac{\mathrm{d} \theta}{\mathrm{dx}} \Rightarrow \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{1}{2} \sec ^{2}\left(\frac{\theta}{2}\right) \times \frac{\mathrm{d} \theta}{\mathrm{dx}}\)
\(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{1}{2} \sec ^{2}\left(\frac{\theta}{2}\right) \times \frac{1}{\mathrm{a} \sin \theta}\)
\(\frac{d^{2} y}{d x^{2}}=\frac{1}{2 a \cos ^{2}\left(\frac{\theta}{2}\right)} \times \frac{1}{\sin \theta} \Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{\operatorname{cosec} \theta}{2 a \cos ^{2}\left(\frac{\theta}{2}\right)}\)