If $x=2 \sin \theta$ and $y=2 \cos \theta$, then the value of $\frac{d^2 y}{d x^2}$ at $\theta=0$ is
Answer & explanation
Correct answer: option 1
$x=2 \sin \theta ~~~y=2 \cos \theta$
so $\frac{d x}{d \theta}=2 \cos \theta, \frac{d y}{d \theta}=-2\sin \theta$
$\Rightarrow \frac{d y / d \theta}{d x / d \theta}=\frac{-2 \sin \theta}{2 \cos \theta}=-\tan \theta$
so $\frac{d^2 y}{d x^2}=-\sec ^2 \theta \frac{d \theta}{d x}$
$\Rightarrow \frac{d^2 y}{d x^2}=\frac{-\sec ^2 \theta}{2 \cos \theta}$
$\Rightarrow\left.\frac{d^2 y}{d x^2}\right]_{\theta=0}=\frac{-1}{2}$
Option: 1