The equation of the parabola with vertex at (0, 0) and focus at (0, 4) is

Options

A. `y^(2) =16x`
B. `x^(2) = 16y`
C. `x^(2) = 4y`
D. `y^(2) = 4x`

Answer

To find the equation of the parabola with a vertex at (0, 0) and a focus at (0, 4), we can follow these steps:

### Step 1: Identify the vertex and focus
The vertex of the parabola is given as (0, 0) and the focus is at (0, 4).

### Step 2: Determine the orientation of the parabola
Since the focus is above the vertex (0, 0), the parabola opens upwards.

### Step 3: Use the standard form of the equation of a parabola
The standard form of the equation of a parabola that opens upwards is given by:
\[ x^2 = 4ay \]
where \( a \) is the distance from the vertex to the focus.

### Step 4: Calculate the value of \( a \)
The distance \( a \) can be calculated as the distance from the vertex (0, 0) to the focus (0, 4):
\[ a = 4 \]

### Step 5: Substitute \( a \) into the standard equation
Now, substituting \( a = 4 \) into the standard equation:
\[ x^2 = 4(4)y \]
\[ x^2 = 16y \]

### Step 6: Write the final equation
Thus, the equation of the parabola is:
\[ x^2 = 16y \]

### Conclusion
The equation of the parabola with vertex at (0, 0) and focus at (0, 4) is:
\[ x^2 = 16y \]

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