If the points `(a_1,b_1),(a_2,b_2)` and `(a_1+a_2,b_1+b_2)` are collinear, then:

Options

A. `a_1 b_2=a_2 b_1`
B. `a_1 + a_2 = b_1 + b_2`
C. `a_1 a_2= b_1 b_2`
D. `a_1 + b_1 = a_2 + b_2`

Answer

To determine the condition for the points \((a_1, b_1)\), \((a_2, b_2)\), and \((a_1 + a_2, b_1 + b_2)\) to be collinear, we can use the concept of the area of a triangle formed by these three points. If the area is zero, then the points are collinear.

### Step-by-Step Solution:

1. **Set Up the Area Formula**:
The area \(A\) of the triangle formed by the points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the determinant:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For our points, we have:
- \((x_1, y_1) = (a_1, b_1)\)
- \((x_2, y_2) = (a_2, b_2)\)
- \((x_3, y_3) = (a_1 + a_2, b_1 + b_2)\)

2. **Substituting the Points into the Area Formula**:
We substitute the coordinates into the area formula:
\[ A = \frac{1}{2} \left| a_1(b_2 - (b_1 + b_2)) + a_2((b_1 + b_2) - b_1) + (a_1 + a_2)(b_1 - b_2) \right| \]
This simplifies to:
\[ A = \frac{1}{2} \left| a_1(-b_1) + a_2(b_2) + (a_1 + a_2)(b_1 - b_2) \right| \]

3. **Simplifying the Expression**:
Expanding the expression:
\[ A = \frac{1}{2} \left| -a_1b_1 + a_2b_2 + a_1b_1 - a_1b_2 + a_2b_1 - a_2b_2 \right| \]
This reduces to:
\[ A = \frac{1}{2} \left| a_2b_1 - a_1b_2 \right| \]

4. **Setting the Area to Zero**:
For the points to be collinear, the area must be zero:
\[ \frac{1}{2} \left| a_2b_1 - a_1b_2 \right| = 0 \]
This implies:
\[ a_2b_1 - a_1b_2 = 0 \]

5. **Final Condition**:
Rearranging gives us the condition:
\[ a_2b_1 = a_1b_2 \]

### Conclusion:
The points \((a_1, b_1)\), \((a_2, b_2)\), and \((a_1 + a_2, b_1 + b_2)\) are collinear if and only if:
\[ a_2b_1 = a_1b_2 \]