What is the equation of parabola whose vertex is at (0, 0) and focus is at (0, -2)?

Options

A. `y^(2)+8x=0`
B. `y^(2)-8x=0`
C. `x^(2)+8y=0`
D. `x^(2)-8y=0`

Answer

To find the equation of the parabola with a vertex at (0, 0) and a focus at (0, -2), 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, -2). This means that the parabola opens downwards since the focus is below the vertex.

### Step 2: Determine the value of 'p'
The distance 'p' from the vertex to the focus is calculated as follows:
- The vertex is at (0, 0) and the focus is at (0, -2).
- Therefore, \( p = -2 - 0 = -2 \).

### Step 3: Write the standard form of the parabola
The standard form of a parabola that opens downwards (with vertex at the origin) is given by the equation:
\[ x^2 = -4py \]
Substituting the value of \( p \):
\[ x^2 = -4(-2)y \]
\[ x^2 = 8y \]

### Step 4: Rearranging the equation
To express the equation in a standard form:
\[ x^2 - 8y = 0 \]

Thus, the equation of the parabola is:
\[ x^2 = 8y \]

### Final Answer
The equation of the parabola whose vertex is at (0, 0) and focus is at (0, -2) is:
\[ x^2 = 8y \]

---