Length of the focal chord of the parabola `(y +3)^(2) = -8(x-1)` which lies at a distance 2 units from the vertex of the parabola is (a) `8` (b) `6sqrt(2)` (c) 9 (d) `5sqrt(3)`

Options

A. 8
B. `6sqrt(2)`
C. 9
D. `5sqrt(3)`

Explanation

To solve the problem of finding the length of the focal chord of the parabola \((y + 3)^2 = -8(x - 1)\) that lies at a distance of 2 units from the vertex, we can follow these steps:

### Step 1: Identify the standard form of the parabola
The given equation of the parabola is \((y + 3)^2 = -8(x - 1)\). This can be rewritten in the standard form of a parabola that opens to the left:
\[ (y - k)^2 = -4p(x - h) \]
where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus.

### Step 2: Determine the vertex and parameters
From the equation \((y + 3)^2 = -8(x - 1)\), we can identify:
- Vertex \((h, k) = (1, -3)\)
- The value of \(-4p = -8\), thus \(4p = 8\) which gives \(p = 2\).

### Step 3: Find the focus
The focus of the parabola is located at a distance \(p\) from the vertex. Since the parabola opens to the left, the focus will be:
\[ \text{Focus} = (h - p, k) = (1 - 2, -3) = (-1, -3) \]

### Step 4: Determine the length of the focal chord
The length of the focal chord is given by the formula:
\[ \text{Length of the focal chord} = 4p \]
Substituting the value of \(p\):
\[ \text{Length of the focal chord} = 4 \times 2 = 8 \]

### Conclusion
Thus, the length of the focal chord of the parabola that lies at a distance of 2 units from the vertex is \(8\).

### Final Answer
The answer is \(\boxed{8}\).