An equation of the parabola whose focus is `(-3,0)` and the directrix is x +5=0 is :

Options

A. `y^(2)=4(x+5)`
B. `y^(2)=4(x+4)`
C. `y^(2)=4(x+3)`
D. `y^(2)=4(x-3)`

Answer

To find the equation of the parabola with focus at (-3, 0) and directrix given by the line x + 5 = 0, 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 as the line \( x + 5 = 0 \), which simplifies to \( x = -5 \).

### Step 2: Determine the vertex of the parabola
The vertex of the parabola lies halfway between the focus and the directrix. The x-coordinate of the vertex can be calculated as the average of the x-coordinates of the focus and the directrix:
\[ x_v = \frac{-3 + (-5)}{2} = \frac{-8}{2} = -4 \]
The y-coordinate of the vertex is the same as that of the focus since the directrix is vertical:
\[ y_v = 0 \]
Thus, the vertex \( V \) is at \( (-4, 0) \).

### Step 3: Determine the distance from the vertex to the focus
The distance \( p \) from the vertex to the focus is:
\[ p = -3 - (-4) = 1 \]
Since the focus is to the left of the vertex, this indicates that the parabola opens to the left.

### Step 4: Write the standard form of the parabola
The standard form of a parabola that opens to the left is given by:
\[ (y - k)^2 = -4p(x - h) \]
where \( (h, k) \) is the vertex. Substituting \( h = -4 \), \( k = 0 \), and \( p = 1 \):
\[ (y - 0)^2 = -4(1)(x + 4) \]
This simplifies to:
\[ y^2 = -4(x + 4) \]

### Step 5: Final equation of the parabola
Thus, the equation of the parabola is:
\[ y^2 = -4(x + 4) \]