The area of the triangle formed by the lines joining the focus of the parabola `y^(2) = 12x` to the points on it which have abscissa 12 are

Options

A. (a) 100 sq. units
B. (b) 130 sq. units
C. (c) 120 sq. units
D. (d) 108 sq. units

Answer

To find the area of the triangle formed by the lines joining the focus of the parabola \(y^2 = 12x\) to the points on it which have abscissa 12, we can follow these steps:

### Step 1: Identify the focus of the parabola
The given parabola is \(y^2 = 12x\). We can compare this with the standard form of a parabola \(y^2 = 4ax\). Here, \(4a = 12\), which gives us \(a = 3\).

The focus of the parabola is located at the point \((a, 0)\), which is \((3, 0)\).

### Step 2: Find the points on the parabola with abscissa 12
To find the points on the parabola where the abscissa (x-coordinate) is 12, we substitute \(x = 12\) into the equation of the parabola:

\[ y^2 = 12 \cdot 12 = 144 \]

Taking the square root, we find:

\[ y = 12 \quad \text{and} \quad y = -12 \]

Thus, the two points on the parabola are \(A(12, 12)\) and \(B(12, -12)\).

### Step 3: Calculate the lengths of the sides of the triangle
We have the focus \(S(3, 0)\) and the points \(A(12, 12)\) and \(B(12, -12)\).

1. **Distance \(SA\)**:
\[ SA = \sqrt{(12 - 3)^2 + (12 - 0)^2} = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \]

2. **Distance \(SB\)**:
\[ SB = \sqrt{(12 - 3)^2 + (-12 - 0)^2} = \sqrt{9^2 + (-12)^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \]

3. **Distance \(AB\)**:
\[ AB = \sqrt{(12 - 12)^2 + (12 - (-12))^2} = \sqrt{0 + 24^2} = 24 \]

### Step 4: Calculate the area of triangle \(SAB\)
To find the area of triangle \(SAB\), we can use the formula:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

In this case, we can take \(AB\) as the base and the height will be the perpendicular distance from point \(S\) to line \(AB\).

The height can be calculated as the distance from the focus \(S(3, 0)\) to the line \(y = 12\) (the line containing points \(A\) and \(B\)):

\[ \text{Height} = 12 - 0 = 12 \]

Thus, the area of triangle \(SAB\) is:

\[ \text{Area} = \frac{1}{2} \times 24 \times 12 = 144 \text{ square units} \]

### Step 5: Adjust for symmetry
Since the parabola is symmetric about the x-axis, the area of triangle \(SAB\) is the same as the area of triangle \(SAB'\) (where \(B'\) is the reflection of \(B\) across the x-axis). Therefore, the total area of triangle \(ASB\) is:

\[ \text{Total Area} = 2 \times \text{Area of } SAB = 2 \times 144 = 288 \text{ square units} \]

However, we only need the area of triangle \(SAB\) formed by the lines joining the focus to points on the parabola.

### Final Result
The area of the triangle formed by the lines joining the focus of the parabola to the points on it which have abscissa 12 is:

\[ \text{Area} = 108 \text{ square units} \]