Question

The length of tangent to the curve $x=a(\cos t+\log \left(\tan \frac{t}{2}\right)),y=a\sin t,$ at any point is :

• A. $\left|a\right|$
• B. $\left|ax\right|$
• C. $\left|ay\right|$
• D. $xy$

Answer 1

The correct option is A $\left|a\right|$

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
But, $\frac{dx}{dt}=a(-\sin t+\frac{1}{2}.\frac{1}{\tan \frac{t}{2}}\cdot {\sec }^2\frac{t}{2})$
$=a(-\sin t+\frac{1}{\sin t})=\frac{a{\cos }^{2}t}{\sin t}$
and $\frac{dy}{dt}=a\cos t$
$\therefore \frac{dy}{dx}=\frac{\left(a\cos t\right)\sin t}{a{\cos }^{2}t}=\tan t$
$\{\because \text{length of tangent}=\left|\frac{y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}}{\left(\frac{dy}{dx}\right)}\right|\}$
$=\left|\frac{a\sin t\sqrt{1+{\tan }^{2}t}}{\tan t}\right|$
$=\left|\frac{a\sin t\cdot \sec t}{\tan t}\right|=\left|a\right|$