If points (a, 0), (0, b) and (1, 1) are collinear, then $\left( \frac{a+b}{ab} \right)$ =

Question

Accepted Answer

To determine why Option 3 is the correct answer, we need to analyze the condition for collinearity of the points (a, 0), (0, b), and (1, 1).

### Explanation:

1. **Collinearity Condition**:
   - Three points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are collinear if the area of the triangle formed by these points is zero. The area can be calculated using the determinant formula:
     $$
     \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|
     $$
   - For points $(a, 0)$, $(0, b)$, and $(1, 1)$, the determinant becomes:
     $$
     \text{Area} = \frac{1}{2} \left| a(b-1) + 0(1-0) + 1(0-b) \right| = \frac{1}{2} \left| ab - a - b \right|
     $$
   - Setting the area to zero for collinearity:
     $$
     ab - a - b = 0
     $$

2. **Solving the Equation**:
   - Rearrange the equation:
     $$
     ab - a - b = 0 \implies ab = a + b
     $$
   - Divide both sides by $ab$ (assuming $a, b \neq 0$):
     $$
     1 = \frac{a}{ab} + \frac{b}{ab} = \frac{1}{b} + \frac{1}{a}
     $$
   - This simplifies to:
     $$
     \frac{a+b}{ab} = 1
     $$

3. **Conclusion**:
   - The expression $\left( \frac{a+b}{ab} \right)$ evaluates to 1, which corresponds to Option 3.

### Why Other Options are Incorrect:

- **Option 1 (-1)**: The equation derived from the collinearity condition does not yield $-1$. The simplification clearly shows that the result is 1, not $-1$.

- **Option 2 ($\sqrt{2}$)**: There is no involvement of square roots in the simplification process. The algebraic manipulation strictly leads to a rational number, specifically 1.

- **Option 4 (2)**: The simplification of the collinearity condition does not result in 2. The derived equation $\frac{a+b}{ab} = 1$ clearly shows that the result is 1.

Thus, the correct answer is indeed Option 3, as the mathematical derivation supports this conclusion.