Question

If $x=asin\theta$ and $y=bcos\theta$, then $\frac{d^{2}y}{d{x}^2}$ is equal to

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

Answer 1

Answer: (D)

Given that, $x=asin\theta$ and $y=bcos\theta$
On differentiating $w.r.t$. $\theta$, we get
$\frac{dx}{d\theta }=acos\theta$ and $\frac{dy}{d\theta }=-bsin\theta$
$\therefore \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }$
$=\frac{-b}{a}tan\theta$
Again differentiating $w.r.t.x$, we get
$\frac{d^{2}y}{d{x}^2}=\frac{-b}{a}se{c}^2\theta \frac{d\theta }{dx}$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=-\frac{b}{a}se{c}^2\theta \left(\frac{1}{acos\theta }\right)$
$=-\frac{b}{a^2}se{c}^3\theta$