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 can follow these steps:

### Step 1: Identify the standard form of the parabola
The given equation can be rewritten in the standard form of a parabola that opens downwards:
\[ (x - h)^{2} = -4a(y - k) \]
Here, \((h, k)\) is the vertex, and \(4a\) is the coefficient of \((y - k)\).

### Step 2: Find the vertex, focus, and latus rectum
From the equation \((x+2)^{2} = -12(y-1)\):
- The vertex \((h, k)\) is \((-2, 1)\).
- The value of \(4a = 12\), hence \(a = 3\).
- The focus is located at \((h, k - a) = (-2, 1 - 3) = (-2, -2)\).
- The endpoints of the latus rectum are given by \((h \pm 2a, k - a)\). Thus, the endpoints are:
- \((-2 + 6, -2) = (4, -2)\)
- \((-2 - 6, -2) = (-8, -2)\)

### Step 3: Choose one endpoint of the latus rectum
We can choose the endpoint \((4, -2)\) for our triangle.

### Step 4: Calculate the area of the triangle
The triangle is formed by the points:
- Vertex \((-2, 1)\)
- Focus \((-2, -2)\)
- Endpoint of latus rectum \((4, -2)\)

To find the area of the triangle, we can use the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the distance between the vertex and the focus, and the height is the horizontal distance from the vertex to the endpoint of the latus rectum.

### Step 5: Calculate the base and height
- The base (distance between vertex and focus):
\[ \text{Base} = |k - (k - a)| = |1 - (-2)| = 3 \]
- The height (horizontal distance from vertex to endpoint):
\[ \text{Height} = |h - 4| = |-2 - 4| = 6 \]

### Step 6: Substitute into the area formula
Now we can substitute the base and height into the area formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 6 = \frac{18}{2} = 9 \]

### Final Answer
The area of the triangle formed by the vertex, focus, and one end of the latus rectum is \(9\) square units.

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