Question

For the curve $x=a(\cos t+\log \left(\tan \frac{t}{2}\right)),y=a\sin t$, the length of the subtangent is

• A. $|a\tan t|$
• B. $|a\cot t|$
• C. $|a|$
• D. $|a\cos t|$

Answer 1

The correct option is D $|a\cos t|$

Given, $y=a\sin t$

$\Rightarrow \frac{dy}{dt}=a\cos t$ .....(1)

Also given, $x=a(\cos t+\log \tan \frac{t}{2})$

$\Rightarrow \frac{dx}{dt}=a(-\sin t+\frac{1}{2}\frac{{\sec }^2\frac{t}{2}}{\tan \frac{t}{2}})$

$\Rightarrow {\frac{dx}{dt}}^{\prime }=a(-\sin t+\frac{1}{\sin t})$

$\Rightarrow \frac{dx}{dt}=a\left(\frac{-{\sin }^{2}t+1}{\sin t}\right)$

$\Rightarrow \frac{dx}{dt}=\frac{a{\cos }^{2}t}{\sin t}$ ---- Equation 2

Now, $\frac{dx}{dy}=\frac{\frac{dx}{dt}}{\frac{dy}{dt}}$

$\Rightarrow \frac{dx}{dy}=\frac{\frac{a{\cos }^{2}t}{\sin t}}{a\cos t}$ (Using (1) and (2))

$\Rightarrow \frac{dx}{dy}=\frac{\cos t}{\sin t}$

Length of subtangent $=|y\frac{dx}{dy}|$

$=|a\sin t\times \frac{\cos t}{\sin t}|$

$=|a\cos t|$