If x = 3 tan t and y = 3 sec t, then the value of


If $x=3 \tan t$ and $y=3 \sec t$, then the value of

$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ at $\mathrm{t}=\frac{\pi}{4}$, is:

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

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

  3. $\frac{1}{6}$

  4. $\frac{1}{6 \sqrt{2}}$

Correct Option: , 4


$\frac{\mathrm{dx}}{\mathrm{dt}}=3 \sec ^{2} \mathrm{t}$

$\frac{\mathrm{dy}}{\mathrm{dt}}=3 \sec t \tan t$

$\frac{d y}{d x}=\frac{\tan t}{\sec t}=\sin t$

$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\cos \mathrm{t} \frac{\mathrm{dt}}{\mathrm{dx}}$

$=\frac{\cos t}{3 \sec ^{2} t}=\frac{\cos ^{3} t}{3}=\frac{1}{3.2 \sqrt{2}}=\frac{1}{6 \sqrt{2}}$

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