Question

The length of the sub tangent at ${}^{\prime }{t}^{\prime }$ on the curve $x=a(t+\sin t),y=a(1-\cos t)$ is

• A. $2a{\sin }^3\left(\frac{t}{2})\sec (\frac{t}{2}\right)$
• B. $a\sin t$
• C. $2a\sin \left(\frac{t}{2}\right)\tan \left(\frac{t}{2}\right)$
• D. $2a\sin \frac{t}{2}$

Answer 1

Answer: (B)

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 subtangent $=\frac{y}{dy/dx}$
$=\frac{a(1-\cos t)}{\tan \frac{t}{2}}$
$=2a\sin \frac{t}{2}\cos \frac{t}{2}$
$=a\sin t$