The equation of parabola with focus at `(-3,0)` and directrix `x +3 = 0` is

Options

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

Answer

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 \( S \) is given as \( (-3, 0) \) and the directrix is given by the equation \( x + 3 = 0 \), which simplifies to \( x = -3 \).

### Step 2: Define a point on the parabola
Let \( P(x, y) \) be any point on the parabola.

### Step 3: Use the definition of a parabola
According to the definition of a parabola, the distance from any point \( P \) on the parabola to the focus \( S \) is equal to the distance from \( P \) to the directrix. Therefore, we can set up the equation:

\[ \text{Distance from } P \text{ to } S = \text{Distance from } P \text{ to the directrix} \]

### Step 4: Calculate the distances
1. The distance from \( P(x, y) \) to the focus \( S(-3, 0) \) is given by:
\[ PS = \sqrt{(x + 3)^2 + (y - 0)^2} = \sqrt{(x + 3)^2 + y^2} \]

2. The distance from \( P(x, y) \) to the directrix \( x = -3 \) is simply the horizontal distance:
\[ PD = |x + 3| \]

### Step 5: Set the distances equal
Since these distances are equal, we have:
\[ \sqrt{(x + 3)^2 + y^2} = |x + 3| \]

### Step 6: Square both sides
To eliminate the square root, we square both sides:
\[ (x + 3)^2 + y^2 = (x + 3)^2 \]

### Step 7: Simplify the equation
Subtract \( (x + 3)^2 \) from both sides:
\[ y^2 = 0 \]

### Step 8: Rearrange the equation
This indicates that the parabola opens along the x-axis. We can express the equation in the standard form of a parabola:
\[ y^2 = 4px \]
where \( p \) is the distance from the vertex to the focus.

### Step 9: Identify the vertex and value of \( p \)
The vertex of the parabola is halfway between the focus and the directrix. The focus is at \( (-3, 0) \) and the directrix is at \( x = -3 \). The vertex is at the midpoint, which is \( (0, 0) \). The distance \( p \) from the vertex to the focus is \( 3 \).

### Step 10: Write the final equation
Since the parabola opens to the right, we have:
\[ y^2 = 12x \]

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

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