If x=2 at,y=at^2, where a is a constant, then d^2y dx^2 at x=1 2 … - Vidyadip

MCQ

If $\text{x}=2\text{ at},\text{y}=\text{at}^2,$ where a is a constant, then $\frac{\text{d}^2\text{y}}{\text{dx}^2}\text{ at}\ \text{x}=\frac{1}{2}$ is:

✓
$\frac{1}{2}\text{a}$
B
1
C
2a
D
None of these

✓

Answer

Correct option: A.

$\frac{1}{2}\text{a}$

Here,

$\text{x}=2\text{ at},\text{y}=\text{at}^2,$

Differentiating w.r.t.x, we get

$\frac{\text{dx}}{\text{dt}}=2\text{a}\ \text{and}\ \frac{\text{dy}}{\text{dt}}=2\text{at}$

$\therefore\frac{\text{dy}}{\text{dx}}=\frac{2\text{at}}{2\text{a}}=\text{t}$

Differentiating w.r.t.x, we get

$\frac{\text{d}^2\text{y}}{\text{dx}^2}=1\times\frac{\text{dt}}{\text{dx}}=\frac{1}{2\text{a}}$

Now, $\Big[\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big]_{\text{x}=\frac{1}{2}}=\frac{1}{2\text{a}}$

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