Question

If $x=2cost-cos2t$ , $y=2sint-sin2t$ , then the value of $\frac{d^{2}y}{d{x}^2}{|}_{t=\pi /2}$ is

• A. $3/2$
• B. $5/2$
• C. $4/2$
• D. $-3/2$

Answer 1

Answer: (D)

$\frac{dx}{dt}=-2\sin t+2\sin 2t$
$\frac{dy}{dt}=2\cos t-2\cos 2t$
$\therefore \frac{dy}{dx}=\frac{2\cos t-2\cos 2t}{-2\sin t+2\sin 2t}$
$=\frac{\cos t-\cos 2t}{\sin 2t-\sin t}$
$=\frac{2\sin \frac{3t}{2}\cdot \sin \frac{t}{2}}{2\cos \frac{3t}{2}\cdot \sin \frac{t}{2}}$
$=\tan \frac{3t}{2}$
$\frac{d^{2}y}{d{x}^2}={\sec }^2\frac{3t}{2}\cdot \frac{3}{2}\cdot \frac{dt}{dx}$
$=\frac{3}{2}{\sec }^2\frac{3t}{2}\frac{1}{(2\sin 2t-2\sin t)}$
$\Rightarrow {\frac{d^{2}y}{d{x}^2}|}_{t=\pi /2}=-3/2$