If ST and SN are the lengths of the subtangen | vedclass

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(A) Given the parametric equations of the curve: $x = a(	heta + \sin 	heta)$ and $y = a(1 - \cos 	heta)$.First,we find the derivatives with respect

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$ST = SN$

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(A) Given the parametric equations of the curve: $x = a(	heta + \sin 	heta)$ and $y = a(1 - \cos 	heta)$.First,we find the derivatives with respect to $	heta$:$\frac{dx}{d	heta} = a(1 + \cos 	heta)$$\frac{dy}{d	heta} = a\sin 	heta$Now,calculate the slope of the tangent $\frac{dy}{dx}$ at $	heta = \frac{\pi}{2}$:$\frac{dy}{dx} = \frac{dy/d	heta}{dx/d	heta} = \frac{a\sin 	heta}{a(1 + \cos 	heta)} = \frac{\sin 	heta}{1 + \cos 	heta}$At $	heta = \frac{\pi}{2}$,$\frac{dy}{dx} = \frac{\sin(\pi/2)}{1 + \cos(\pi/2)} = \frac{1}{1 + 0} = 1$.The value of $y$ at $	heta = \frac{\pi}{2}$ is $y = a(1 - \cos(\pi/2)) = a(1 - 0) = a$.The length of the subtangent $ST$ is given by $\left| \frac{y}{dy/dx} ight| = \left| \frac{a}{1} ight| = a$.The length of the subnormal $SN$ is given by $|y \cdot \frac{dy}{dx}| = |a \cdot 1| = a$.Since $ST = a$ and $SN = a$,we conclude that $ST = SN$.

If $ST$ and $SN$ are the lengths of the subtangent and the subnormal at the point $\theta = \frac{\pi}{2}$ on the curve $x = a(\theta + \sin \theta ), y = a(1 - \cos \theta ), a \neq 0$,then:

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