A) \[a\,\,\sin \,t\]
B) \[2a\,\,\sin \,\left( \frac{t}{2} \right)\tan \left( \frac{t}{2} \right)\]
C) \[2a\,\,\sin \,\frac{t}{2}\]
D) \[2a\,\,{{\sin }^{3}}\,\left( \frac{t}{2} \right)\sec \left( \frac{t}{2} \right)\]
Correct Answer: A
Solution :
Given, \[x=a(t+\sin t),y=a(1-\cos t)\] \[\Rightarrow \] \[\frac{dx}{dt}=a(1+\cos t),\frac{dy}{dt}=a(\sin t)\] \[\therefore \] \[\frac{dy}{dx}=\frac{a\sin t}{a(1+\cos t)}=\tan \frac{t}{2}\] \[\therefore \] Length of sub tangent \[=\frac{y}{dy/dx}\] \[\frac{a(1-\cos \,t)}{\tan \frac{1}{2}}\] \[=2a\,\sin \frac{t}{2}\cos \frac{t}{2}\] \[=a\,\sin t\]
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