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. 1
B. `2sqrt2`
C. `sqrt2`
D. 2

Answer

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

### Step 1: Understand the problem
We need to find the sum of the x-intercept (a) and the 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: Use the intercept form of the line
The equation of a line in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \(a\) is the x-intercept and \(b\) is the y-intercept.

### Step 3: Calculate the area enclosed by the line and the axes
The area \(A\) of the triangle formed by the line and the coordinate axes is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times b \]
According to the problem, this area is equal to 4 square units:
\[ \frac{1}{2} \times a \times b = 4 \]
Multiplying both sides by 2, we get:
\[ a \times b = 8 \quad \text{(Equation 1)} \]

### Step 4: Substitute the point (2, 2) into the line equation
Since the line passes through the point (2, 2), we can substitute \(x = 2\) and \(y = 2\) into the intercept form equation:
\[ \frac{2}{a} + \frac{2}{b} = 1 \]
Factoring out 2 gives us:
\[ \frac{2}{a} + \frac{2}{b} = 1 \implies \frac{1}{a} + \frac{1}{b} = \frac{1}{2} \]
Taking the least common multiple, we can rewrite this as:
\[ \frac{b + a}{ab} = \frac{1}{2} \]
Cross-multiplying gives us:
\[ 2(a + b) = ab \quad \text{(Equation 2)} \]

### Step 5: Solve the equations
Now we have two equations:
1. \(ab = 8\) (from Equation 1)
2. \(2(a + b) = ab\) (from Equation 2)

Substituting \(ab = 8\) into Equation 2:
\[ 2(a + b) = 8 \]
Dividing both sides by 2:
\[ a + b = 4 \]

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