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

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)`

Answer

To find the equation of the parabola with the vertex at the point \((-3, 0)\) and the directrix given by the line \(x + 5 = 0\), we can follow these steps:

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

### Step 2: Determine the Orientation of the Parabola
Since the directrix is a vertical line and is to the left of the vertex, the parabola opens to the right.

### Step 3: Calculate the Distance \(a\)
The distance \(a\) between the vertex and the directrix can be calculated as follows:
- The vertex is at \(x = -3\).
- The directrix is at \(x = -5\).
- The distance \(a\) is given by:
\[ a = \text{Distance from vertex to directrix} = |-3 - (-5)| = |-3 + 5| = |2| = 2 \]

### Step 4: Write the Standard Form of the Parabola
The standard form of a parabola that opens to the right is given by:
\[ (y - k)^2 = 4a(x - h) \]
where \((h, k)\) is the vertex of the parabola.

### Step 5: Substitute the Values
Here, \(h = -3\), \(k = 0\), and \(a = 2\). Substituting these values into the equation gives:
\[ (y - 0)^2 = 4 \cdot 2 \cdot (x - (-3)) \]
This simplifies to:
\[ y^2 = 8(x + 3) \]

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