If the points (a,b),(c,d) and (a-c,b-d) are collinear, then

Options

A. `ab=cd`
B. `ac=bd`
C. `ad=bc`
D. None

Answer

To determine the condition for the points (a, b), (c, d), and (a - c, b - d) to be collinear, we can use the concept of the area of a triangle formed by these points. If the area is zero, the points are collinear.

### Step-by-Step Solution:

1. **Identify the Points**:
Let the points be:
- Point A: (a, b)
- Point B: (c, d)
- Point C: (a - c, b - d)

2. **Area of Triangle Formula**:
The area \( A \) 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 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, the area can be expressed as:
\[ A = \frac{1}{2} \begin{vmatrix} a & b & 1 \\ c & d & 1 \\ a - c & b - d & 1 \end{vmatrix} \]

3. **Set the Area to Zero**:
Since the points are collinear, we set the area to zero:
\[ \frac{1}{2} \begin{vmatrix} a & b & 1 \\ c & d & 1 \\ a - c & b - d & 1 \end{vmatrix} = 0 \]

4. **Calculate the Determinant**:
Expanding the determinant, we have:
\[ \begin{vmatrix} a & b & 1 \\ c & d & 1 \\ a - c & b - d & 1 \end{vmatrix} = a(d - (b - d)) + c((b - d) - b) + (a - c)(b - d - d) \]
Simplifying this gives:
\[ = a(d - b + d) + c(b - d - b) + (a - c)(b - 2d) \]
This simplifies to:
\[ = ad - ab + ad - cb + ab - 2cd + ac - bc \]
Rearranging leads to:
\[ = 2ad - ab - bc - 2cd = 0 \]

5. **Final Condition**:
Rearranging the terms gives us:
\[ ad = bc \]

### Conclusion:
Thus, the condition for the points (a, b), (c, d), and (a - c, b - d) to be collinear is:
\[ ad = bc \]