If `x=a t^2,\ \ y=2\ a t` , then `(d^2y)/(dx^2)=` `-1/(t^2)` (b) `1/(2\ a t^3)` (c) `-1/(t^3)` (d) `-1/(2\ a t^3)`

Answer

To find \(\frac{d^2y}{dx^2}\) given the equations \(x = at^2\) and \(y = 2at\), we will follow these steps:

### Step 1: Differentiate \(x\) with respect to \(t\)
Given:
\[ x = at^2 \]
Differentiating both sides with respect to \(t\):
\[ \frac{dx}{dt} = \frac{d}{dt}(at^2) = 2at \]

### Step 2: Differentiate \(y\) with respect to \(t\)
Given:
\[ y = 2at \]
Differentiating both sides with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d}{dt}(2at) = 2a \]

### Step 3: Find \(\frac{dy}{dx}\)
Using the chain rule, we can express \(\frac{dy}{dx}\) as:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the values from Steps 1 and 2:
\[ \frac{dy}{dx} = \frac{2a}{2at} = \frac{1}{t} \]

### Step 4: Differentiate \(\frac{dy}{dx}\) with respect to \(t\)
Now, we differentiate \(\frac{dy}{dx} = \frac{1}{t}\) with respect to \(t\):
\[ \frac{d}{dt}\left(\frac{1}{t}\right) = -\frac{1}{t^2} \]

### Step 5: Find \(\frac{dx}{dt}\) and its reciprocal
From Step 1, we have:
\[ \frac{dx}{dt} = 2at \]
Thus, the reciprocal is:
\[ \frac{dt}{dx} = \frac{1}{2at} \]

### Step 6: Use the chain rule to find \(\frac{d^2y}{dx^2}\)
Now we can find \(\frac{d^2y}{dx^2}\) using the formula:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx} \]
Substituting the values we found:
\[ \frac{d^2y}{dx^2} = \left(-\frac{1}{t^2}\right) \cdot \left(\frac{1}{2at}\right) = -\frac{1}{2at^3} \]

### Final Answer
Thus, we have:
\[ \frac{d^2y}{dx^2} = -\frac{1}{2at^3} \]