If (a, b), (c, d) and (a - c, b - d) are collinear, then which one of the following is correct ?

Options

A. bc - ad = 0
B. ab - cd = 0
C. bc + ad = 0
D. ab + cd = 0

Answer

To determine the relationship between the points (a, b), (c, d), and (a - c, b - d) when they are collinear, we can use the concept of the area of a triangle formed by these points. If the area is zero, it implies that the points are collinear.

### Step-by-Step Solution:

1. **Set up the points**: We have three points:
- Point 1: \( P_1 = (a, b) \)
- Point 2: \( P_2 = (c, d) \)
- Point 3: \( P_3 = (a - c, b - d) \)

2. **Formulate the determinant**: The area of the triangle formed by these three points can be expressed using the determinant of a matrix. The determinant for the points can be set up as follows:
\[ \text{Area} = \frac{1}{2} \begin{vmatrix} a & b & 1 \\ c & d & 1 \\ a - c & b - d & 1 \end{vmatrix} \]

3. **Calculate the determinant**: We need to calculate the determinant:
\[ \begin{vmatrix} a & b & 1 \\ c & d & 1 \\ a - c & b - d & 1 \end{vmatrix} \]

4. **Perform row operations**: To simplify the determinant, we can perform row operations:
- Replace \( R_2 \) with \( R_2 - R_1 \) (i.e., \( R_2' = (c - a, d - b, 0) \))
- Replace \( R_3 \) with \( R_3 - R_1 \) (i.e., \( R_3' = (0, 0, 0) \))

The determinant now becomes:
\[ \begin{vmatrix} a & b & 1 \\ c - a & d - b & 0 \\ 0 & 0 & 0 \end{vmatrix} \]

5. **Evaluate the determinant**: The determinant of a matrix with a row of zeros is zero:
\[ \text{Determinant} = 0 \]

6. **Conclusion**: Since the determinant is zero, this implies that the points are collinear. Therefore, we can conclude that the relationship between \( a, b, c, d \) must satisfy the equation:
\[ a(d - b) + b(c - a) = 0 \]
This is the condition for collinearity of the three points.

### Final Answer:
The correct relationship derived from the collinearity condition is given by:
\[ ad - bc = 0 \]