If the points (1, 1) , (a, 0) and (0, b) are collinear then, what is the value of a + b ?

Options

A. 0
B. 1
C. ab
D. None of these

Answer

To find the value of \( a + b \) given that the points \( (1, 1) \), \( (a, 0) \), and \( (0, b) \) are collinear, we can use the concept that the area of the triangle formed by these three points is zero.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( (x_1, y_1) = (1, 1) \)
- Let \( (x_2, y_2) = (a, 0) \)
- Let \( (x_3, y_3) = (0, b) \)

2. **Use the Area Formula**:
The area \( A \) of a triangle formed by three points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For the points given, we substitute:
\[ A = \frac{1}{2} \left| 1(0 - b) + a(b - 1) + 0(1 - 0) \right| \]

3. **Simplify the Expression**:
\[ A = \frac{1}{2} \left| -b + ab - a \right| \]
\[ A = \frac{1}{2} \left| ab - a - b \right| \]

4. **Set the Area to Zero**:
Since the points are collinear, the area is zero:
\[ \frac{1}{2} \left| ab - a - b \right| = 0 \]
This implies:
\[ ab - a - b = 0 \]

5. **Rearranging the Equation**:
Rearranging gives us:
\[ ab = a + b \]

6. **Finding \( a + b \)**:
We can express this as:
\[ a + b - ab = 0 \]
This means:
\[ a + b = ab \]

7. **Conclusion**:
The value of \( a + b \) can be expressed in terms of \( a \) and \( b \) as \( a + b = ab \).

### Final Answer:
The value of \( a + b \) is \( ab \).