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 of finding the value of \( a + b - ab \) given that the points \( (a, 0) \), \( (0, b) \), and \( (1, 1) \) are collinear, we can follow these steps:

### Step 1: Understand the condition of collinearity
Three points are collinear if the slopes between any two pairs of points are equal. We will calculate the slopes between the points \( (a, 0) \) and \( (1, 1) \), and between \( (0, b) \) and \( (1, 1) \).

### Step 2: Calculate the slope between points \( (a, 0) \) and \( (1, 1) \)
Using the slope formula:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
For points \( (a, 0) \) and \( (1, 1) \):
- \( (x_1, y_1) = (a, 0) \)
- \( (x_2, y_2) = (1, 1) \)

The slope is:
\[ \text{slope}_{AC} = \frac{1 - 0}{1 - a} = \frac{1}{1 - a} \]

### Step 3: Calculate the slope between points \( (0, b) \) and \( (1, 1) \)
For points \( (0, b) \) and \( (1, 1) \):
- \( (x_1, y_1) = (0, b) \)
- \( (x_2, y_2) = (1, 1) \)

The slope is:
\[ \text{slope}_{BC} = \frac{1 - b}{1 - 0} = 1 - b \]

### Step 4: Set the slopes equal to each other
Since the points are collinear, we set the two slopes equal:
\[ \frac{1}{1 - a} = 1 - b \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 1 = (1 - b)(1 - a) \]

### Step 6: Expand the equation
Expanding the right side:
\[ 1 = 1 - a - b + ab \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ 0 = -a - b + ab \]
or
\[ ab - a - b = 0 \]

### Step 8: Rearranging further
This can be rearranged to:
\[ ab - a - b + 1 = 1 \]

### Step 9: Factor the left side
This can be factored as:
\[ (a - 1)(b - 1) = 1 \]

### Step 10: Find \( a + b - ab \)
We want to find \( a + b - ab \). From our rearranged equation:
\[ ab - a - b = 0 \implies a + b - ab = -1 \]

Thus, the value of \( a + b - ab \) is:
\[ \boxed{-1} \]