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

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understand Collinearity**:
For three points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) to be collinear, the area of the triangle formed by these points must be zero. The area can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Setting the area to zero gives us the condition for collinearity.

**Hint**: Remember that collinear points lie on the same straight line, which means the area formed by them is zero.

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

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

**Hint**: The absolute value can be ignored since we are setting it to zero.

4. **Set the Expression to Zero**:
From the equation \( ab - a - b = 0 \), we can rearrange it:
\[ ab = a + b \]

**Hint**: This equation can be factored to find a relationship between \( a \) and \( b \).

5. **Rearranging the Equation**:
Rearranging gives:
\[ ab - a - b + 1 = 1 \]
Factoring this, we have:
\[ (a - 1)(b - 1) = 1 \]

**Hint**: This product indicates that \( a \) and \( b \) are related to the number 1.

6. **Finding \( \frac{1}{a} + \frac{1}{b} \)**:
We can express \( \frac{1}{a} + \frac{1}{b} \) as:
\[ \frac{1}{a} + \frac{1}{b} = \frac{b + a}{ab} \]
From our previous result \( ab = a + b \), we substitute:
\[ \frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab} = \frac{ab}{ab} = 1 \]

**Hint**: This shows that the sum of the reciprocals of \( a \) and \( b \) simplifies nicely.

7. **Conclusion**:
Therefore, the value of \( \frac{1}{a} + \frac{1}{b} \) is \( 1 \).

**Final Answer**: \( \frac{1}{a} + \frac{1}{b} = 1 \)