Point A lies on the line `y=2x` and the sum of its abscissa and ordinate is 12. Point `B` lies on the x-axis and the line `AB` is perpendicular to the line `y=2x`. Let `O` be the origin. The area of the Delta AOB is

Options

A. 20
B. 40
C. 60
D. 80

Explanation

To solve the problem, we will follow these steps:

### Step 1: Determine the coordinates of point A
Given that point A lies on the line \( y = 2x \) and the sum of its abscissa (x-coordinate) and ordinate (y-coordinate) is 12, we can denote the coordinates of point A as \( (x_1, y_1) \).

From the information provided:
1. \( y_1 = 2x_1 \) (since point A lies on the line \( y = 2x \))
2. \( x_1 + y_1 = 12 \)

Substituting the first equation into the second:
\[ x_1 + 2x_1 = 12 \\ 3x_1 = 12 \\ x_1 = 4 \]

Now substituting \( x_1 \) back to find \( y_1 \):
\[ y_1 = 2x_1 = 2 \times 4 = 8 \]

Thus, the coordinates of point A are \( (4, 8) \).

### Step 2: Find the coordinates of point B
Point B lies on the x-axis, so its coordinates can be represented as \( (b, 0) \). The line AB is perpendicular to the line \( y = 2x \).

The slope of the line \( y = 2x \) is 2. The slope of line AB, which is perpendicular to it, can be calculated using the negative reciprocal:
\[ m_{AB} = -\frac{1}{2} \]

Using the point-slope form of the equation of a line, we can write the equation of line AB:
\[ y - y_1 = m_{AB}(x - x_1) \\ y - 8 = -\frac{1}{2}(x - 4) \]

Now, substituting \( y = 0 \) to find the x-coordinate of point B:
\[ 0 - 8 = -\frac{1}{2}(b - 4) \\ -8 = -\frac{1}{2}(b - 4) \\ 16 = b - 4 \\ b = 20 \]

Thus, the coordinates of point B are \( (20, 0) \).

### Step 3: Calculate the area of triangle AOB
The area \( A \) of triangle AOB can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base OB is the distance along the x-axis from O to B, which is 20 units, and the height is the y-coordinate of point A, which is 8 units.

Calculating the area:
\[ \text{Area} = \frac{1}{2} \times 20 \times 8 = \frac{160}{2} = 80 \]

Thus, the area of triangle AOB is \( 80 \).

### Final Answer
The area of triangle AOB is \( 80 \).

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