If the points (a, 0), (0, b) and (1, 1) are collinear, then

Options

A. `(1)/(a^(2)) + (1)/(b^(2)) = 1`
B. `(1)/(a^(2)) - (1)/(b^(2)) = 1`
C. `(1)/(a) + (1)/(b) = 1`
D. `(1)/(a) - (1)/(b) = 1`

Answer

To determine the relationship between the points (a, 0), (0, b), and (1, 1) when they are collinear, we can use the concept of the area of a triangle formed by these points. If the area is zero, the points are collinear.

### Step-by-Step Solution:

1. **Identify the Points**: We have three points:
- Point 1: \( (a, 0) \)
- Point 2: \( (0, b) \)
- Point 3: \( (1, 1) \)

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

3. **Substituting Values into the Formula**:
\[ A = \frac{1}{2} \left| a(b - 1) + 0(1 - 0) + 1(0 - b) \right| \]
Simplifying this, we get:
\[ A = \frac{1}{2} \left| ab - a - b \right| \]

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

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

6. **Dividing by \( ab \)** (assuming \( a \) and \( b \) are not zero):
\[ 1 = \frac{a}{b} + \frac{b}{a} \]
Rearranging gives:
\[ \frac{1}{a} + \frac{1}{b} = 1 \]

7. **Conclusion**: Thus, we find that:
\[ \frac{1}{a} + \frac{1}{b} = 1 \]
This corresponds to option C: \( \frac{1}{A} + \frac{1}{B} = 1 \).