Question

If $x=a \cos ^4 \theta, y=a \sin ^4 \theta$, then $\frac{d y}{d x}$ at $\theta=\frac{3 \pi}{4}$ is

(A) -1

(B) 1

(C) $-a^2$

(D) $a^2$

Answer 1

On differentiating w.r.t. $\theta$, we get

$\begin{array}{l}\text { } \quad \begin{aligned}\frac{d x}{d \theta} & =4 a \cos ^3 \theta(-\sin \theta) \\\frac{d y}{d \theta} & =4 a \sin ^3 \theta \cos \theta \\\frac{d y}{d x} & =\frac{d y / d \theta}{d x / d \theta}=-\frac{4 a \sin ^3 \theta \cos \theta}{4 a \cos ^3 \theta \sin \theta} \\& =-\tan ^2 \theta \\\therefore \quad\left(\frac{d y}{d x}\right)_{\theta=\frac{3 \pi}{4}} & =-\tan ^2\left(\frac{3 \pi}{4}\right)=-1\end{aligned}\end{array}$

So, The correct option is (A).