The equation of the parabola with focus (3, 0) and directrix y = -3 is

Options

A. `x^(2) = 12y`
B. `x^(2) = -12y`
C. `y^(2) = 12x`
D. `y^(2) = -12x`

Answer

To find the equation of the parabola with focus at (3, 0) and directrix y = -3, we can follow these steps:

### Step 1: Identify the coordinates of the focus and the equation of the directrix
- The focus is given as (3, 0).
- The directrix is given as the line y = -3.

### Step 2: Determine the vertex of the parabola
The vertex of a parabola is located halfway between the focus and the directrix.

- The y-coordinate of the focus is 0, and the y-coordinate of the directrix is -3.
- The midpoint (vertex) can be calculated as:
\[ y_{vertex} = \frac{0 + (-3)}{2} = \frac{-3}{2} = -1.5 \]
- The x-coordinate of the vertex is the same as the x-coordinate of the focus, which is 3.
- Therefore, the vertex is at (3, -1.5).

### Step 3: Determine the value of 'p'
The distance 'p' is the distance from the vertex to the focus (or from the vertex to the directrix).

- The distance from the vertex (3, -1.5) to the focus (3, 0) is:
\[ p = 0 - (-1.5) = 1.5 \]

### Step 4: Write the equation of the parabola
Since the parabola opens upwards (the focus is above the directrix), the standard form of the equation of a parabola is given by:
\[ (x - h)^2 = 4p(y - k) \]
where (h, k) is the vertex.

- Here, \( h = 3 \), \( k = -1.5 \), and \( p = 1.5 \).
- Plugging these values into the equation:
\[ (x - 3)^2 = 4 \cdot 1.5 \cdot (y + 1.5) \]
\[ (x - 3)^2 = 6(y + 1.5) \]

### Step 5: Simplify the equation
Expanding the equation:
\[ (x - 3)^2 = 6y + 9 \]
This can be rearranged to:
\[ (x - 3)^2 - 9 = 6y \]
Thus, the final equation of the parabola is:
\[ 6y = (x - 3)^2 - 9 \]

### Final Equation
The equation of the parabola is:
\[ y = \frac{(x - 3)^2 - 9}{6} \]

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