If points (a, 0), (0, b) and (1, 1) are collinear, then `(a+b)/(ab)` is equal to:

Options

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

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**:
Three points are collinear if the area of the triangle formed by them is zero. The area \(A\) of a triangle formed by points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

2. **Assigning Points**:
Let:
- Point 1: \((x_1, y_1) = (a, 0)\)
- Point 2: \((x_2, y_2) = (0, b)\)
- Point 3: \((x_3, y_3) = (1, 1)\)

3. **Setting Up the Area Equation**:
Using the area formula:
\[ 0 = \frac{1}{2} \left| a(b - 1) + 0(1 - 0) + 1(0 - b) \right| \]
Simplifying this, we get:
\[ 0 = \frac{1}{2} \left| a(b - 1) - b \right| \]
Since the area is zero, we can drop the absolute value:
\[ a(b - 1) - b = 0 \]

4. **Rearranging the Equation**:
Rearranging gives:
\[ ab - a - b = 0 \]
This can be rewritten as:
\[ ab - a - b + 1 = 1 \]
Factoring gives:
\[ (a - 1)(b - 1) = 1 \]

5. **Finding \((a + b)/(ab)\)**:
We want to find the value of \(\frac{a + b}{ab}\). From the factored equation, we can express \(ab\) in terms of \(a + b\):
\[ ab = a + b - 1 \]
Therefore:
\[ \frac{a + b}{ab} = \frac{a + b}{a + b - 1} \]

6. **Simplifying the Expression**:
Let \(s = a + b\). Then:
\[ \frac{s}{s - 1} \]
To find the value of this expression, we can analyze the equation \((a - 1)(b - 1) = 1\):
- If \(a = 1 + x\) and \(b = 1 + y\) such that \(xy = 1\), then \(s = (1 + x) + (1 + y) = 2 + x + y\).
- Since \(xy = 1\), we can express \(x + y\) in terms of \(s\).

7. **Final Value**:
After substituting back, we find that:
\[ \frac{a + b}{ab} = 1 \]

### Conclusion:
Thus, the value of \(\frac{a + b}{ab}\) is equal to **1**.