Question

At any point $t$ on the curve $x=a(t+\sin t),y=a(1-\cot )$, find the length of the tangent, normal subtangent and subnormal .

Answer 1

$x=a(t+\sin t)\Rightarrow \frac{dx}{dt}=a.\cos t$

$y=a(1-\cot )=\frac{dy}{dt}=a.\sin t$

$\frac{dy}{dx}=\tan t$

length of tangent

$\frac{y_1\sqrt{1+f^{{}^{\prime }}(x_1{)}^2}}{f^{{}^{\prime }}\left(x_1\right)}$

$=\frac{a(1-\cos t)\sqrt{(}1+{\tan }^{2}t)}{\tan t}$

length of sub normal

$=\frac{y_1}{f^{{}^{\prime }}\left(x_1\right)}$

$=\frac{a(1-\cos t)}{\tan t}$