The area of the triangle formed by the lines joining the vertex of the parabola `x^(2) = 12 y` to the ends of its latus rectum is

Options

A. 12 sq. units
B. 16 sq. units
C. 18 sq.units
D. 24 sq. units

Answer

To find the area of the triangle formed by the lines joining the vertex of the parabola \( x^2 = 12y \) to the ends of its latus rectum, we can follow these steps:

### Step 1: Identify the parameters of the parabola
The given parabola is \( x^2 = 12y \). We can rewrite it in the standard form \( x^2 = 4ay \) to identify the value of \( a \).

### Step 2: Compare with the standard form
From the equation \( x^2 = 12y \), we can see that \( 4a = 12 \). Therefore, we find:
\[ a = \frac{12}{4} = 3 \]

### Step 3: Determine the vertex and the latus rectum
The vertex of the parabola is at the point \( (0, 0) \). The latus rectum of the parabola is a horizontal line at \( y = a \), which gives us:
\[ y = 3 \]

### Step 4: Find the endpoints of the latus rectum
To find the endpoints of the latus rectum, we substitute \( y = 3 \) into the parabola's equation:
\[ x^2 = 12 \cdot 3 = 36 \]
Taking the square root, we find:
\[ x = \pm 6 \]
Thus, the endpoints of the latus rectum are \( (6, 3) \) and \( (-6, 3) \).

### Step 5: Identify the coordinates of the triangle's vertices
The vertices of the triangle formed by the vertex of the parabola and the endpoints of the latus rectum are:
1. Vertex: \( (0, 0) \)
2. Endpoint 1: \( (6, 3) \)
3. Endpoint 2: \( (-6, 3) \)

### Step 6: Use the formula for the area of a 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:
- \( (x_1, y_1) = (0, 0) \)
- \( (x_2, y_2) = (6, 3) \)
- \( (x_3, y_3) = (-6, 3) \)

### Step 7: Calculate the area
Substituting the values into the area formula:
\[ A = \frac{1}{2} \left| 0(3 - 3) + 6(3 - 0) + (-6)(0 - 3) \right| \]
This simplifies to:
\[ A = \frac{1}{2} \left| 0 + 18 + 18 \right| = \frac{1}{2} \left| 36 \right| = \frac{36}{2} = 18 \]

### Conclusion
The area of the triangle formed by the lines joining the vertex of the parabola to the ends of its latus rectum is:
\[ \text{Area} = 18 \text{ square units} \]