Question

The length of the sub tangent at $'t'$ 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, $x=a(t+\sin t), y=a(1-\cos t)$
$\Rightarrow \frac{d x}{d t}=a(1+\cos t), \frac{d y}{d t}=a(\sin t)$
$\therefore \frac{d y}{d x}=\frac{a \sin t}{a(1+\cos t)}=\tan \frac{t}{2}$
$\therefore $ Length of subtangent $=\frac{y}{d y / d x}$
$=\frac{a(1-\cos t)}{\tan \frac{t}{2}}$
$=2 a \sin \frac{t}{2} \cos \frac{t}{2}$
$=a \sin t$