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If \(x\) and \(y\) are connected parametrically by the equations given in $ x = a ( \cos \theta + \theta \sin \theta), y = a ( \sin \theta - \theta \cos \: \theta) $ without eliminating the parameter, Find \(\frac{\large dy}{\large dx}\).

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  • By chain rule we have $\large\frac{dy}{dx}=\frac{dy}{d\theta}$$\times\large\frac{d\theta}{dx}$
Step 1:
$x=a(\cos\theta+\theta\sin\theta)$
Differentiating with respect to $\theta$
$\large\frac{dx}{d\theta}$$=a(-\sin\theta+\sin\theta+\theta\cos\theta)$
$\quad\;\;=a\theta\cos\theta$
Step 2:
$y=a(\sin\theta-\theta\cos\theta)$
Differentiating with respect to $\theta$
$\large\frac{dy}{d\theta}$$=\cos\theta-1.\cos\theta-\theta.(-\sin\theta)$
$\quad\;\;=a(\theta\sin\theta)$
Step 3:
$\large\frac{dy}{dx}=\frac{dy}{d\theta}$$\times\large\frac{d\theta}{dx}$
$\quad\;\;=a.\theta\sin\theta\large\frac{1}{a\theta\cos\theta}$
$\quad\;\;=\large\frac{\sin\theta}{\cos\theta}$
$\quad\;\;=\tan\theta$
answered May 10, 2013 by sreemathi.v
 

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