A line passing through the point (2, 2) encloses an area of 4 sq. units with coordinate axes. The sum of intercepts made by the line on the x and y axis is equal to

Options

A. `-2`
B. 4
C. `-4`
D. 2

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the problem
We need to find the sum of the x-intercept (A) and y-intercept (B) of a line that passes through the point (2, 2) and encloses an area of 4 square units with the coordinate axes.

### Step 2: Write the equation of the line
The equation of a line with x-intercept A and y-intercept B can be written as:
\[ \frac{x}{A} + \frac{y}{B} = 1 \]

### Step 3: Substitute the point (2, 2)
Since the line passes through the point (2, 2), we can substitute \(x = 2\) and \(y = 2\) into the equation:
\[ \frac{2}{A} + \frac{2}{B} = 1 \]

### Step 4: Rearranging the equation
Multiplying through by \(AB\) to eliminate the denominators gives:
\[ 2B + 2A = AB \]
Rearranging this, we have:
\[ AB - 2A - 2B = 0 \]
This can be factored as:
\[ (A - 2)(B - 2) = 4 \]

### Step 5: Area of the triangle
The area \(A\) of the triangle formed by the line and the axes is given by:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times A \times B \]
We know this area is equal to 4 square units, so:
\[ \frac{1}{2} \times A \times B = 4 \]
Multiplying both sides by 2 gives:
\[ A \times B = 8 \]

### Step 6: Solve the equations
We now have two equations:
1. \(AB = 8\)
2. \((A - 2)(B - 2) = 4\)

Expanding the second equation:
\[ AB - 2A - 2B + 4 = 4 \]
Substituting \(AB = 8\) into this gives:
\[ 8 - 2A - 2B + 4 = 4 \]
Simplifying this results in:
\[ -2A - 2B + 12 = 4 \]
\[ -2A - 2B = -8 \]
Dividing through by -2 gives:
\[ A + B = 4 \]

### Conclusion
Thus, the sum of the intercepts made by the line on the x-axis and y-axis is:
\[ \boxed{4} \]