If the vertex of a parabola is the point `(-3,0)` and the directrix is the line `x+5=0` , then find its equation.

Options

A. `y^(2)=8(x+3)`
B. `x^(2)=8(y+3)`
C. `y^(2)=-8(x+3)`
D. `y^(2)=8(x+5)`

Explanation

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

### Step 1: Identify the vertex and directrix
The vertex of the parabola is given as \( V(-3, 0) \) and the directrix is given by the equation \( x + 5 = 0 \), which simplifies to \( x = -5 \).

### Step 2: Determine the focus
The focus of the parabola lies on the axis of symmetry, which is perpendicular to the directrix. The vertex is the midpoint between the focus and the directrix. Since the directrix is at \( x = -5 \) and the vertex is at \( x = -3 \), we can find the focus \( F \).

The distance from the vertex to the directrix is:
\[ d = |-3 - (-5)| = 2 \]
Since the focus lies to the right of the vertex (because the parabola opens to the right), the x-coordinate of the focus will be:
\[ -3 + 2 = -1 \]
Thus, the coordinates of the focus are \( F(-1, 0) \).

### Step 3: Use the definition of a parabola
A parabola is defined as the set of all points \( P(x, y) \) such that the distance from \( P \) to the focus \( F \) is equal to the distance from \( P \) to the directrix.

The distance from point \( P(x, y) \) to the focus \( F(-1, 0) \) is given by:
\[ d_F = \sqrt{(x + 1)^2 + (y - 0)^2} = \sqrt{(x + 1)^2 + y^2} \]

The distance from point \( P(x, y) \) to the directrix \( x = -5 \) is:
\[ d_D = |x + 5| \]

### Step 4: Set the distances equal
According to the definition of a parabola, we set these two distances equal:
\[ \sqrt{(x + 1)^2 + y^2} = |x + 5| \]

### Step 5: Square both sides
To eliminate the square root, we square both sides:
\[ (x + 1)^2 + y^2 = (x + 5)^2 \]

### Step 6: Expand both sides
Expanding both sides gives:
\[ (x^2 + 2x + 1 + y^2) = (x^2 + 10x + 25) \]

### Step 7: Simplify the equation
Now, we can simplify the equation:
\[ x^2 + 2x + 1 + y^2 = x^2 + 10x + 25 \]
Subtract \( x^2 \) from both sides:
\[ 2x + 1 + y^2 = 10x + 25 \]
Rearranging gives:
\[ y^2 = 10x - 2x + 25 - 1 \]
\[ y^2 = 8x + 24 \]

### Step 8: Rearranging the equation
We can write this in standard form:
\[ y^2 = 8(x + 3) \]

### Final Equation
Thus, the equation of the parabola is:
\[ y^2 = 8(x + 3) \]