The equation of the parabola with `(-3, 0) ` and focus and `x+5=0` as directrix, is

Options

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

Answer

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

### Step 1: Identify the focus and directrix
The focus of the parabola is given as \((-3, 0)\) and the directrix is the line \(x = -5\).

### Step 2: Use the definition of a parabola
A parabola is defined as the set of all points \(P(h, k)\) such that the distance from \(P\) to the focus is equal to the distance from \(P\) to the directrix.

### Step 3: Calculate the distance from the point \(P(h, k)\) to the focus
The distance from \(P(h, k)\) to the focus \((-3, 0)\) is given by:
\[ d_{focus} = \sqrt{(h + 3)^2 + (k - 0)^2} = \sqrt{(h + 3)^2 + k^2} \]

### Step 4: Calculate the distance from the point \(P(h, k)\) to the directrix
The directrix is the line \(x = -5\). The distance from point \(P(h, k)\) to the directrix is:
\[ d_{directrix} = |h + 5| \]

### Step 5: Set the distances equal
According to the definition of a parabola, we set the distances equal:
\[ \sqrt{(h + 3)^2 + k^2} = |h + 5| \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (h + 3)^2 + k^2 = (h + 5)^2 \]

### Step 7: Expand both sides
Expanding both sides, we have:
\[ (h^2 + 6h + 9 + k^2) = (h^2 + 10h + 25) \]

### Step 8: Simplify the equation
Now, we can simplify the equation:
\[ h^2 + 6h + 9 + k^2 = h^2 + 10h + 25 \]
Subtract \(h^2\) from both sides:
\[ 6h + 9 + k^2 = 10h + 25 \]
Rearranging gives:
\[ k^2 = 10h - 6h + 25 - 9 \]
\[ k^2 = 4h + 16 \]

### Step 9: Write the equation in standard form
We can rewrite this as:
\[ k^2 = 4(h + 4) \]
In terms of \(x\) and \(y\), we replace \(h\) with \(x\) and \(k\) with \(y\):
\[ y^2 = 4(x + 4) \]

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