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 \( a \) and \( b \) for the points \( A(1,2) \), \( B(0,0) \), and \( C(a,b) \) to be collinear, we can use the area of the triangle formed by these points. If the points are collinear, the area of the triangle will be zero.

### Step-by-Step Solution:

1. **Formula for Area of Triangle**:
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:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

2. **Substituting the Points**:
Let \( A(1, 2) \), \( B(0, 0) \), and \( C(a, b) \). We can substitute these coordinates into the area formula:
\[ A = \frac{1}{2} \left| 1(0 - b) + 0(b - 2) + a(2 - 0) \right| \]

3. **Simplifying the Expression**:
Simplifying the expression inside the absolute value:
\[ A = \frac{1}{2} \left| 1(-b) + 0 + a(2) \right| = \frac{1}{2} \left| -b + 2a \right| \]

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

5. **Removing Absolute Value**:
The absolute value equation \( \left| -b + 2a \right| = 0 \) leads to:
\[ -b + 2a = 0 \]

6. **Rearranging the Equation**:
Rearranging gives us the relationship between \( a \) and \( b \):
\[ b = 2a \]

### Final Answer:
The relationship between \( a \) and \( b \) for the points \( A(1,2) \), \( B(0,0) \), and \( C(a,b) \) to be collinear is:
\[ b = 2a \]