Question

If $x=a\sin \theta$ and $y=b\cos \theta$, then $\frac{d^{2}y}{d{x}^2}$ is

• A. $\frac{a}{b^2}{\sec }^2\theta$
• B. $\frac{-b}{a}{\sec }^2\theta$
• C. $\frac{b}{a^2}{\sec }^3\theta$
• D. $\frac{-b}{a^2}{\sec }^3\theta$

Answer 1

Answer: (D)

$\frac{-b}{a^2}{\sec }^3\theta$

Given, $x=a\sin \theta$ and $y=b\cos \theta$
On differentiating w.r.t.$\theta$, we get
$\frac{dx}{d\theta }=a\cos \theta$
and $\frac{dy}{d\theta }=-b\sin \theta$
$\Rightarrow \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }=-\frac{b}{a}\tan \theta$
Again, differentiating w.r.t. $x$, we get
$\frac{d^{2}y}{d{x}^2}=-\frac{b}{a}se{c}^2\theta \cdot \frac{d\theta }{dx}$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=-\frac{b}{a}se{c}^2\theta \cdot \frac{1}{a\cos \theta }$
$=-\frac{b}{a^2}se{c}^3\theta$