Area of the triangle formed by the vertex, focus and one end of latusrectum of the parabola `(x+2)^(2)=-12(y-1)` is

Options

A. 36
B. 18
C. 9
D. 6

Explanation

To find the area of the triangle formed by the vertex, focus, and one end of the latus rectum of the parabola given by the equation \((x + 2)^2 = -12(y - 1)\), we will follow these steps:

### Step 1: Identify the standard form of the parabola
The given equation can be rewritten as:
\[ (x + 2)^2 = -12(y - 1) \]
This is in the form \((x - h)^2 = -4a(y - k)\), where \((h, k)\) is the vertex of the parabola.

### Step 2: Determine the vertex
From the equation, we can see that:
- \(h = -2\)
- \(k = 1\)

Thus, the vertex \(V\) is at the point:
\[ V(-2, 1) \]

### Step 3: Find the value of \(a\)
From the equation, we have:
\[ -4a = -12 \implies 4a = 12 \implies a = 3 \]

### Step 4: Determine the focus
The focus of the parabola is given by the coordinates \((h, k - a)\):
\[ F(-2, 1 - 3) = F(-2, -2) \]

### Step 5: Find the endpoints of the latus rectum
The endpoints of the latus rectum are located at:
\[ (h - 2a, k - a) \quad \text{and} \quad (h + 2a, k - a) \]
Calculating these points:
- For the left endpoint:
\[ L_1 = (-2 - 2 \cdot 3, 1 - 3) = (-8, -2) \]
- For the right endpoint:
\[ L_2 = (-2 + 2 \cdot 3, 1 - 3) = (4, -2) \]

### Step 6: Identify the points of the triangle
The triangle is formed by the points:
- Vertex \(V(-2, 1)\)
- Focus \(F(-2, -2)\)
- One endpoint of the latus rectum \(L_1(-8, -2)\)

### Step 7: Calculate the area of the triangle
The area \(A\) of a triangle formed by points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates:
- \(V(-2, 1)\) → \((x_1, y_1)\)
- \(F(-2, -2)\) → \((x_2, y_2)\)
- \(L_1(-8, -2)\) → \((x_3, y_3)\)

The area becomes:
\[ A = \frac{1}{2} \left| -2(-2 + 2) + (-2)(-2 - 1) + (-8)(1 + 2) \right| \]
Calculating each term:
\[ = \frac{1}{2} \left| -2(0) + (-2)(-3) + (-8)(3) \right| \]
\[ = \frac{1}{2} \left| 0 + 6 - 24 \right| \]
\[ = \frac{1}{2} \left| -18 \right| = \frac{18}{2} = 9 \]

### Final Answer
The area of the triangle formed by the vertex, focus, and one end of the latus rectum of the parabola is:
\[ \boxed{9} \]