if x = 3 tan t and y = 3 sec t,
Question:

If $x=3 \tan t$ and $y=3 \sec t$, then the value of $\frac{d^{2} y}{d x^{2}}$ at

$\mathrm{t}=\frac{\pi}{4}$, is:

1. (1) $\frac{1}{3 \sqrt{2}}$

2. (2) $\frac{1}{6 \sqrt{2}}$

3. (3) $\frac{3}{2 \sqrt{2}}$

4. (4) $\frac{1}{6}$

Correct Option: 2,

Solution:

$\because \quad x=3 \tan t \Rightarrow \frac{d x}{d t}=3 \sec ^{2} t$

and $y=3 \sec t \Rightarrow \frac{d y}{d t}=3 \sec t \cdot \tan t$

$\because \quad \frac{d y}{d x}=\frac{d y / d t}{d x / d t} \quad \therefore \quad \frac{d y}{d x}=\frac{\tan t}{\sec t}=\sin t$

$\therefore \quad \frac{d^{2} y}{d x^{2}}=\frac{d}{d t}(\sin t) \cdot \frac{d t}{d x}$

$=\cos t \cdot \frac{1}{3 \sec ^{2} t}$

$\therefore \quad \frac{d^{2} y}{d x^{2}}\left(\right.$ at $\left.t=\frac{\pi}{4}\right)=\frac{1}{3} \cdot\left(\frac{1}{\sqrt{2}}\right)^{3}$

$=\frac{1}{6 \sqrt{2}}$