If ` x = at^(2) and y = 2 ` at then

Options

A. `x^(2) = 4ay`
B. `y^(2) = 4ax`
C. ` x^(2) + y^(2) = a^(2)`
D. `x^(2) - y^(2) = a^(2)`

Answer

To find the relationship between \( x \) and \( y \) given the equations \( x = at^2 \) and \( y = 2at \), we can follow these steps:

### Step 1: Write down the equations
We have:
\[ x = at^2 \]
\[ y = 2at \]

### Step 2: Solve for \( a \) in terms of \( y \)
From the equation \( y = 2at \), we can express \( a \) as:
\[ a = \frac{y}{2t} \]

### Step 3: Substitute \( a \) into the equation for \( x \)
Now, substitute \( a \) into the equation for \( x \):
\[ x = a t^2 = \left(\frac{y}{2t}\right) t^2 \]
This simplifies to:
\[ x = \frac{y t}{2} \]

### Step 4: Express \( t \) in terms of \( y \) and \( x \)
Rearranging the equation \( x = \frac{y t}{2} \) gives:
\[ t = \frac{2x}{y} \]

### Step 5: Substitute \( t \) back into the equation for \( y \)
Now substitute \( t \) back into the equation for \( y \):
\[ y = 2a t = 2a \left(\frac{2x}{y}\right) \]
Substituting \( a = \frac{y}{2t} \) gives:
\[ y = 2 \left(\frac{y}{2 \left(\frac{2x}{y}\right)}\right) \left(\frac{2x}{y}\right) \]
This simplifies to:
\[ y^2 = 4ax \]

### Step 6: Final relationship
Thus, we have the final relationship:
\[ y^2 = 4ax \]