If (a , 0) , (0 , b) and (1 , 1) are collinear , what is (a + b -ab) equal to ?

Options

A. 2
B. 1
C. 0
D. `-1`

Answer

To solve the problem, we need to determine the value of \( a + b - ab \) given that the points \( (a, 0) \), \( (0, b) \), and \( (1, 1) \) are collinear.

### Step-by-step Solution:

1. **Understanding Collinearity**:
For three points to be collinear, the area of the triangle formed by these points must be zero. We can use the formula for the area of a triangle given by three points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

2. **Assigning Points**:
Let \( (x_1, y_1) = (a, 0) \), \( (x_2, y_2) = (0, b) \), and \( (x_3, y_3) = (1, 1) \).

3. **Substituting into the Area Formula**:
Substitute the coordinates into the area formula:
\[ \text{Area} = \frac{1}{2} \left| a(b - 1) + 0(1 - 0) + 1(0 - b) \right| \]
Simplifying this gives:
\[ \text{Area} = \frac{1}{2} \left| ab - a - b \right| \]

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

5. **Rearranging the Equation**:
Rearranging gives us:
\[ ab - a - b + 1 = 1 \]
or
\[ a + b - ab = 0 \]

6. **Finding the Value**:
Thus, we find:
\[ a + b - ab = 0 \]

### Conclusion:
The value of \( a + b - ab \) is equal to \( 0 \).