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, we need to find the sum of the x-intercept and y-intercept of a line that passes through the point (2, 2) and encloses an area of 4 square units with the coordinate axes.

### Step-by-Step Solution:

1. **Understanding the Area Enclosed by the Line:**
The area \( A \) of a triangle formed by the line with the coordinate axes can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the x-intercept \( a \) and the height is the y-intercept \( b \). Given that the area is 4 square units:
\[ \frac{1}{2} \times a \times b = 4 \]
Therefore, we can write:
\[ a \times b = 8 \quad \text{(Equation 1)} \]

2. **Using the Point (2, 2):**
The line passes through the point (2, 2). The equation of the line in intercept form is:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
Substituting the point (2, 2) into the equation gives:
\[ \frac{2}{a} + \frac{2}{b} = 1 \]
Multiplying through by \( ab \) to eliminate the denominators:
\[ 2b + 2a = ab \quad \text{(Equation 2)} \]

3. **Rearranging Equation 2:**
Rearranging Equation 2 gives:
\[ ab - 2a - 2b = 0 \]
This can be factored as:
\[ (a - 2)(b - 2) = 4 \quad \text{(Equation 3)} \]

4. **Solving the System of Equations:**
Now we have two equations:
- \( ab = 8 \) (from Equation 1)
- \( (a - 2)(b - 2) = 4 \) (from Equation 3)

Expanding Equation 3:
\[ ab - 2a - 2b + 4 = 4 \]
Substituting \( ab = 8 \) into the equation:
\[ 8 - 2a - 2b + 4 = 4 \]
Simplifying this gives:
\[ -2a - 2b + 12 = 4 \]
Thus:
\[ -2a - 2b = -8 \quad \Rightarrow \quad a + b = 4 \quad \text{(Equation 4)} \]

5. **Finding the Sum of Intercepts:**
Now we have:
- \( a + b = 4 \)
- \( ab = 8 \)

The sum of the intercepts \( a + b \) is:
\[ a + b = 4 \]

### Final Answer:
The sum of the intercepts made by the line on the x-axis and y-axis is **4**.