To determine the equation of the parabola that opens to the right with a given vertex and focal diameter, follow these steps:
### Step 1: Understand the standard form of the parabola
When a parabola opens to the right (horizontally), its equation is generally given by:
[tex]\[ (y - k)^2 = 4p(x - h) \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola and [tex]\(p\)[/tex] is the distance from the vertex to the focus.
### Step 2: Identify the vertex and the focal diameter
We are given:
- Vertex [tex]\((h, k) = (-9, -4)\)[/tex]
- Focal diameter = 16
### Step 3: Find [tex]\(p\)[/tex]
The focal diameter of the parabola is the length of the chord passing through the focus and perpendicular to the axis of symmetry. This is also equal to [tex]\(|4p|\)[/tex]. To find [tex]\(p\)[/tex], we use the given focal diameter:
[tex]\[ |4p| = 16 \][/tex]
[tex]\[ 4p = 16 \][/tex]
[tex]\[ p = \frac{16}{4} = 4 \][/tex]
### Step 4: Plug the known values into the standard form
Now that we have [tex]\(h = -9\)[/tex], [tex]\(k = -4\)[/tex], and [tex]\(p = 4\)[/tex], we substitute these values into the standard form of the equation of the parabola:
[tex]\[ (y - (-4))^2 = 4 \cdot 4(x - (-9)) \][/tex]
[tex]\[ (y + 4)^2 = 16(x + 9) \][/tex]
### Step 5: Write down the final equation
Hence, the equation of the parabola is:
[tex]\[ (y + 4)^2 = 16(x + 9) \][/tex]
This is the equation of the parabola that opens to the right, has a vertex at [tex]\((-9, -4)\)[/tex], and a focal diameter of 16.
