The equation of a parabola having focus (3, 0) and directrix x + 3 = 0, is

Options

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

Explanation

To find the equation of the parabola with the given focus and directrix, we can follow these steps:

### Step 1: Identify the Focus and Directrix
The focus of the parabola is given as \( F(3, 0) \) and the directrix is given by the equation \( x + 3 = 0 \), which simplifies to \( x = -3 \).

### Step 2: Determine the Vertex
The vertex of the parabola lies halfway between the focus and the directrix. The x-coordinate of the vertex can be found by averaging the x-coordinates of the focus and the line of the directrix.

\[ \text{Vertex} (h, k) = \left( \frac{x_{focus} + x_{directrix}}{2}, y_{focus} \right) \]
\[ h = \frac{3 + (-3)}{2} = \frac{0}{2} = 0 \]
\[ k = 0 \]
Thus, the vertex is at \( (0, 0) \).

### Step 3: Determine the Orientation of the Parabola
Since the focus is to the right of the directrix, the parabola opens to the right.

### Step 4: Find the Distance \( p \)
The distance \( p \) from the vertex to the focus can be calculated as follows:
\[ p = x_{focus} - h = 3 - 0 = 3 \]

### Step 5: Write the Standard Form of the Parabola
The standard form of a parabola that opens to the right is given by:
\[ (y - k)^2 = 4p(x - h) \]
Substituting the values of \( h \), \( k \), and \( p \):
\[ (y - 0)^2 = 4 \cdot 3 (x - 0) \]
This simplifies to:
\[ y^2 = 12x \]

### Final Answer
The equation of the parabola is:
\[ y^2 = 12x \]

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