Answer :
To find the standard form of the equation of the parabola, we will follow these steps:
1. Determine the vertex.
2. Find the value of \( p \), the distance from the vertex to the focus or directrix.
3. Write the equation of the parabola using the standard form for a vertical parabola.
### Step 1: Determine the Vertex
The vertex of a parabola is the midpoint between the focus and the directrix. Given the focus is at (0, 0) and the directrix is the line \( y = 16 \), the vertex has the same x-coordinate as the focus since the parabola is vertical and symmetric with respect to its vertex.
The vertex will thus have an x-coordinate of 0. The y-coordinate of the vertex will be the average of the y-coordinates of the focus and directrix. Since the focus is at y-coordinate 0 and the directrix is at y-coordinate 16, the y-coordinate of the vertex is \( (0 + 16) / 2 = 8 \).
Therefore, the vertex is at (0, 8).
### Step 2: Find the Value of \( p \)
The value of \( p \) is the distance from the vertex to the focus or to the directrix. Since the vertex is at (0, 8) and the focus is at (0, 0), the distance \( p \) is \( 8 - 0 = 8 \).
### Step 3: Write the Equation of the Parabola
The standard form of the equation of a vertical parabola is \( (x - h)^2 = 4p(y - k) \), where \( (h, k) \) is the vertex of the parabola and \( p \) is the distance between the vertex and the focus or directrix.
Since the vertex \( (h, k) \) is at (0, 8), our equation becomes \( (x - 0)^2 = 4p(y - 8) \).
We have determined that \( p = 8 \), so we substitute that into the equation to get \( x^2 = 4 \cdot 8 \cdot (y - 8) \).
Simplify the equation by multiplying out the constants:
\( x^2 = 32(y - 8) \).
Now, the equation is in standard form for a parabola that opens upwards (since \( p \) is positive) with the given focus and directrix.
So the standard form of the equation of the parabola is:
\( x^2 = 32(y - 8) \).
To find the standard form of the equation of a parabola given its focus and directrix, we first need to determine whether the parabola opens upward or downward based on the location of the focus and directrix.
Given:
Focus: (0, 0)
Directrix: y = 16
Since the directrix is above the focus, the parabola opens downward.
The standard form of the equation of a parabola that opens downward is:
\((x - h)^2 = 4p(y - k)\),
where (h, k) is the vertex of the parabola and p is the distance between the vertex and the focus (which is also the distance between the vertex and the directrix for a parabola opening downward).
In this case, the vertex is halfway between the focus and the directrix, so the vertex is \((0, 8)\) (since the focus is at the origin (0, 0) and the directrix is \(y = 16\)).
The distance between the vertex and the focus (or the vertex and the directrix) is 8 units.
Therefore, the standard form of the equation of the parabola is:
\(x^2 = -64y\).
Given:
Focus: (0, 0)
Directrix: y = 16
Since the directrix is above the focus, the parabola opens downward.
The standard form of the equation of a parabola that opens downward is:
\((x - h)^2 = 4p(y - k)\),
where (h, k) is the vertex of the parabola and p is the distance between the vertex and the focus (which is also the distance between the vertex and the directrix for a parabola opening downward).
In this case, the vertex is halfway between the focus and the directrix, so the vertex is \((0, 8)\) (since the focus is at the origin (0, 0) and the directrix is \(y = 16\)).
The distance between the vertex and the focus (or the vertex and the directrix) is 8 units.
Therefore, the standard form of the equation of the parabola is:
\(x^2 = -64y\).
