Question

For $x=a\cos \theta ,y=a\sin \theta$ find $\frac{d^{2}y}{d{x}^2}$ where $\theta \ne K\pi ,K\in Z$.

• A. $\frac{1}{a}{\csc }^3\theta$
• B. ${\sec }^3\theta$
• C. ${\csc }^3\theta$
• D. None of these

Answer 1

Answer: (A)

$\frac{1}{a}{\csc }^3\theta$

$x=a\cos \theta ,y=a\sin \theta$

$x^2+y^2=a^2$

Differentiating w.r.t. x, we get,

$2x+2y\frac{dy}{dx}=\mathrm{0.......}\left(i\right)$

Differentiating w.r.t. x, we get,

$\Rightarrow 2+2{\left(\frac{dy}{dx}\right)}^2+2y\frac{d^{2}y}{d{x}^2}=0$

$\Rightarrow 1+{\left(\frac{dy}{dx}\right)}^2+y\frac{d^{2}y}{d{x}^2}=0$

$\Rightarrow 1+\frac{x^2}{y^2}+y\frac{d^{2}y}{d{x}^2}=0$

$\Rightarrow 1+\frac{{\cos }^2\theta }{{\sin }^2\theta }+a\sin \theta \frac{d^{2}y}{d{x}^2}=0$

$\Rightarrow {\csc }^2\theta +a\sin \theta \frac{d^{2}y}{d{x}^2}=0$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=-\frac{{\csc }^2\theta }{a\sin \theta }$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=-\frac{{\csc }^3\theta }{a}$