Question

The length of normal at $\theta$ on the curve given by $x=a{\cos }^3\theta ,y=a{\sin }^3\theta$ is

• A. $asi{n}^2\theta$
• B. $asi{n}^2\theta |cos\theta |$
• C. $asi{n}^2\theta |tan\theta |$
• D. $a|cos\theta |$

Answer 1

The correct option is C $asi{n}^2\theta |tan\theta |$

Given,

$x=a{\cos }^3\theta$

$\frac{dx}{d\theta }=-3a{\cos }^2\theta \sin \theta$

$y=a{\sin }^3\theta$

$\frac{dy}{d\theta }=3a{\sin }^2\theta \cos \theta$

$\frac{dy}{dx}=\frac{3a{\sin }^2\theta \cos \theta }{-3a{\cos }^2\theta \sin \theta }$

$=-\tan \theta$

Length of normal $=y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$

$=a{\sin }^3\theta \sqrt{1+{\tan }^2\theta }$

$=a{\sin }^3\theta \sqrt{{\sec }^2\theta }$

$=a{\sin }^3\theta \frac{1}{\cos \theta }$

$=a{\sin }^2\theta |\tan \theta |$