Question

If $x=3tant$ and $y=3sect$, then the value of $\frac{d^{2}y}{d{x}^2}$ at $t=\frac{\pi }{4}$, is :

• A. $\frac{3}{2\sqrt{2}}$
• B. $\frac{1}{3\sqrt{2}}$
• C. $\frac{1}{6}$
• D. $\frac{1}{6\sqrt{2}}$

Answer 1

Answer: (D)

$\frac{dx}{dt}=3{\sec }^{2}t$
$\frac{dy}{dt}=3\sec t\tan t$
$\frac{dy}{dx}=\frac{\tan t}{\sec t}=\sin t$
$\frac{d^{2}y}{d{x}^2}=\cos t\frac{dt}{dx}$
$=\frac{\cos t}{3{\sec }^{2}t}=\frac{{\cos }^{3}t}{3}=\frac{1}{3.2\sqrt{2}}=\frac{1}{6\sqrt{2}}$