Question : The length of the normal to the curve \[x=a\left(\theta+\sin{\left(\theta\right)}\right)\ ,\ y=a\ \left(1-\cos{\left(\theta\right)}\right)\] at \[ \theta=\frac{\pi}{2}\] is \[2a\] \[\frac{a}{2}\] \[a\sqrt2\] \[\frac{a}{\sqrt{2\ }}\] Solution : First we shall find derivatives of x and y \[\frac{dx}{d\theta}=a\left(1+\cos{\left(\theta\right)}\right)\] \[\frac{dy}{d\theta}=asin{\left(\theta\right)}\] Now calculating slope \[m=\frac{dy}{dx}|_{\theta=\frac{\pi}{2}}=\frac{asin{\left(\frac{\pi}{2}\right)}}{a\left(1+\cos{\left(\frac{\pi}{2}\right)}\right)}=\frac{1}{1}=1\]…
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