Question

If $x = a \cos ^{4} \theta, y =\operatorname{asin}^{4} \theta$, then $\frac{ dy }{ dx }$ at $\theta=\frac{3 \pi}{4}$ is

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

Answer 1

Answer: (A)

Given that, $x=a \cos ^{4} \theta$ and $y=a \sin ^{4} \theta$
On differentiating w.r.t $\theta$, we get
$\frac{d x}{d \theta}=4 a \cos ^{3} \theta(-\sin \theta)$ and $\frac{d y}{d \theta}=4 a \sin ^{3} \theta \cos \theta$
$\therefore \frac{ dy }{ dx }=\frac{\frac{ dy }{ d \theta}}{\frac{ dx }{ d \theta}}=-\frac{4 a \sin ^{3} \theta \cos \theta}{4 a \cos ^{3} \theta \sin \theta}=-\frac{\sin ^{2} \theta}{\cos ^{2} \theta}=-\tan ^{2} \theta$
Now, $\left(\frac{d y}{d x}\right)_{\theta=\frac{3 \pi}{4}}=-\tan ^{2}\left(\frac{3 \pi}{4}\right)=-1$