For the curve x = a(cos theta + theta sin the | vedclass

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(B) Given the parametric equations: $x = a(\cos 	heta + 	heta \sin 	heta)$ and $y = a(\sin 	heta - 	heta \cos 	heta)$.First,find the derivatives

Vedclass · Accepted Answer

It makes an angle of $(\frac{\pi}{2} + 	heta)$ with the $x$-axis.

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(B) Given the parametric equations: $x = a(\cos 	heta + 	heta \sin 	heta)$ and $y = a(\sin 	heta - 	heta \cos 	heta)$.First,find the derivatives with respect to $	heta$:$\frac{dx}{d	heta} = a(-\sin 	heta + \sin 	heta + 	heta \cos 	heta) = a	heta \cos 	heta$$\frac{dy}{d	heta} = a(\cos 	heta - \cos 	heta + 	heta \sin 	heta) = a	heta \sin 	heta$Now,find the slope of the tangent:$\frac{dy}{dx} = \frac{dy/d	heta}{dx/d	heta} = \frac{a	heta \sin 	heta}{a	heta \cos 	heta} = 	an 	heta$.The slope of the normal is $m_n = -\frac{1}{dy/dx} = -\frac{1}{	an 	heta} = -\cot 	heta$.Since the slope of the normal is $-\cot 	heta = 	an(	heta + \frac{\pi}{2})$,the angle it makes with the $x$-axis is $(\frac{\pi}{2} + 	heta)$.Thus,option $B$ is correct.

For the curve $x = a(\cos \theta + \theta \sin \theta)$,$y = a(\sin \theta - \theta \cos \theta)$,which of the following is true for the normal at any point $\theta$?

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