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 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: Use the Area of Triangle Formula
The area of a triangle formed by three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Since the points are collinear, the area will be equal to 0.

### Step 2: Substitute the Points into the Area Formula
Let’s substitute the points \((a, 0)\), \((0, b)\), and \((1, 1)\) into the area formula:
\[ \text{Area} = \frac{1}{2} \left| a(b - 1) + 0(1 - 0) + 1(0 - b) \right| = 0 \]
This simplifies to:
\[ \frac{1}{2} \left| ab - a - b \right| = 0 \]

### Step 3: Set the Expression Inside the Absolute Value to Zero
Since the area is zero, we can set the expression inside the absolute value to zero:
\[ ab - a - b = 0 \]

### Step 4: Rearrange the Equation
Rearranging gives us:
\[ ab = a + b \]

### Step 5: Divide Both Sides by \(ab\)
Assuming \(ab \neq 0\), we can divide both sides by \(ab\):
\[ 1 = \frac{a}{b} + \frac{b}{a} \]

### Step 6: Rewrite the Equation
This can be rewritten as:
\[ \frac{a + b}{ab} = 1 \]

### Conclusion
Thus, we find that:
\[ \frac{a+b}{ab} = 1 \]

### Final Answer
The value of \(\frac{a+b}{ab}\) is equal to \(1\).

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