Question

If $x=a{\cos }^4\theta ,y=a{\sin }^4\theta$, then $\frac{dy}{dx}$ at $\theta =\frac{\pi }{4}$ is

• A. $-1$
• B. $1$
• C. $-a^2$
• D. $a^2$

Answer 1

The correct option is A $-1$

Given, $x=a{\cos }^4\theta$ and $y=a{\sin }^4\theta$
On differentiating with respect to $\theta$ respectively, we get
$\frac{dx}{d\theta }=4a{\cos }^3\theta (-\sin \theta )$
$=-4a\sin \theta {\cos }^3\theta$
and $\frac{dy}{d\theta }=4a{\sin }^3\theta \cos \theta$
$\therefore \frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}=\frac{4a{\sin }^3\theta \cos \theta }{-4a\sin \theta {\cos }^3\theta }$
$\Rightarrow \frac{dy}{dx}=-\frac{{\sin }^2\theta }{{\cos }^2\theta }=-{\tan }^2\theta$
Now, ${\left(\frac{dy}{dx}\right)}_{\theta =\frac{3\pi }{4}}=-{\tan }^2\left(\frac{3\pi }{4}\right)=-1$