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Question
Find \[\frac{dy}{dx}\] ,When \[x = a \left( 1 - \cos \theta \right) \text{ and } y = a \left( \theta + \sin \theta \right) \text{ at } \theta = \frac{\pi}{2}\] ?
Solution
\[\text{ We have, }x = a\left( 1 - \cos\theta \right) \text{ and }y = a\left( \theta + \sin\theta \right)\]
\[ \therefore \frac{dx}{d\theta} = \frac{d}{d\theta}\left[ a\left( 1 - \cos\theta \right) \right] = a\left( \sin\theta \right)\]
\[\text{ and } \]
\[\frac{dy}{d\theta} = \frac{d}{d\theta}\left[ a\left( \theta + \sin\theta \right) \right] = a\left( 1 + \cos\theta \right)\]
\[ \therefore \left[ \frac{dy}{dx} \right]_{\theta = \frac{\pi}{2}} = \left[ \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \right]_{\theta = \frac{\pi}{2}} = \left[ \frac{a\left( 1 + \cos\theta \right)}{a\left( \sin \theta \right)} \right]_{\theta = \frac{\pi}{2}} = \frac{a\left( 1 + 0 \right)}{a} = 1\]
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