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

Options

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

Explanation

To find 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 will 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, which is \((y - k)^2 = -4a(x - h)\), where \((h, k)\) is the vertex.

### Step 2: Determine the vertex and parameters
From the equation \((y + 3)^2 = -8(x - 1)\):
- The vertex \((h, k)\) is at \((1, -3)\).
- The value of \(4a\) is \(8\), thus \(a = 2\).

### Step 3: Locate the focus and directrix
The focus of the parabola is located at a distance \(a\) from the vertex along the axis of symmetry. Since the parabola opens to the left:
- The focus is at \((h - a, k) = (1 - 2, -3) = (-1, -3)\).
- The directrix is the line \(x = h + a = 1 + 2 = 3\).

### Step 4: Determine the focal chord
A focal chord is a line segment that passes through the focus and is perpendicular to the axis of symmetry. The length of the focal chord can be calculated using the formula \(4a\).

### Step 5: Calculate the length of the focal chord
Since \(a = 2\):
\[ \text{Length of the focal chord} = 4a = 4 \times 2 = 8 \text{ units}. \]

### Conclusion
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 is \(8\) units.

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