A straight line passing through the point`(2,2)` and the axes enclose an area `lamda`. The intercepts on the axes made by the line are given by the two roots of:

Options

A. `x^(2)-2|lambda|x+|lambda|=0`
B. `x^(2)+|lambda|x+2|lambda|=0`
C. `x^(2)-|lambda|x+2|lambda|=0`
D. None of these

Answer

To solve the problem of finding the equation of the straight line passing through the point (2, 2) and determining the intercepts on the axes, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Line Equation**:
The equation of a line in terms of its x-intercept \( A \) and y-intercept \( B \) can be expressed as:
\[ \frac{x}{A} + \frac{y}{B} = 1 \]

2. **Substituting the Point**:
Since the line passes through the point (2, 2), we can substitute \( x = 2 \) and \( y = 2 \) into the line equation:
\[ \frac{2}{A} + \frac{2}{B} = 1 \]

3. **Rearranging the Equation**:
Multiplying through by \( AB \) to eliminate the denominators gives:
\[ 2B + 2A = AB \]
Rearranging this, we have:
\[ AB - 2A - 2B = 0 \]

4. **Factoring the Equation**:
We can rewrite this as:
\[ AB - 2A - 2B + 4 = 4 \]
This can be factored as:
\[ (A - 2)(B - 2) = 4 \]

5. **Finding the Area Enclosed**:
The area \( \lambda \) enclosed by the intercepts on the axes is given by:
\[ \text{Area} = \frac{1}{2} \times A \times B = \lambda \]
Thus, we have:
\[ AB = 2\lambda \]

6. **Sum and Product of Roots**:
From the equation \( AB - 2A - 2B = 0 \), we can express \( A + B \) and \( AB \):
- The sum of the roots \( A + B = 2\lambda \)
- The product of the roots \( AB = 2\lambda \)

7. **Forming the Quadratic Equation**:
The quadratic equation whose roots are \( A \) and \( B \) can be expressed as:
\[ x^2 - (A + B)x + AB = 0 \]
Substituting the values we found:
\[ x^2 - (2\lambda)x + 2\lambda = 0 \]

8. **Final Equation**:
Rearranging gives:
\[ x^2 - 2\lambda x + 2\lambda = 0 \]

### Conclusion:
The correct form of the quadratic equation is:
\[ x^2 - 2\lambda x + 2\lambda = 0 \]
This corresponds to option 3 in the original question.