If the points A(1,2) , B(0,0) and C (a,b) are collinear , then

Options

A. a=b
B. a=2b
C. 2a=b
D. a=-b

Explanation

To determine the relationship between the coordinates of points A(1, 2), B(0, 0), and C(a, b) being collinear, we can use the area of the triangle formed by these points. If the area is zero, the points are collinear.

### Step-by-Step Solution:

1. **Understand the Area Formula**: The area of a triangle formed by three points (x1, y1), (x2, y2), and (x3, y3) is given by the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For points A(1, 2), B(0, 0), and C(a, b), we will use this formula.

2. **Assign Coordinates**: Assign the coordinates:
- A(1, 2) → (x1, y1) = (1, 2)
- B(0, 0) → (x2, y2) = (0, 0)
- C(a, b) → (x3, y3) = (a, b)

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

4. **Set Area to Zero for Collinearity**: Since the points are collinear, the area must equal zero:
\[ \frac{1}{2} \left| -b + 2a \right| = 0 \]
This implies:
\[ -b + 2a = 0 \]

5. **Solve for b**: Rearranging the equation gives:
\[ b = 2a \]

### Conclusion:
Thus, the relationship between b and a is given by:
\[ b = 2a \]