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 coordinates (a, b), (c, d), and (a-c, b-d) being collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Collinearity**:
Three points are collinear if the area of the triangle formed by them is zero. This can be determined using the determinant of the matrix formed by their coordinates.

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

3. **Form the Determinant**:
We can express the condition for collinearity using the determinant:
\[ \begin{vmatrix} a & b & 1 \\ c & d & 1 \\ a-c & b-d & 1 \end{vmatrix} = 0 \]

4. **Calculate the Determinant**:
Expanding the determinant, we have:
\[ = a \begin{vmatrix} d & 1 \\ b-d & 1 \end{vmatrix} - b \begin{vmatrix} c & 1 \\ a-c & 1 \end{vmatrix} + 1 \begin{vmatrix} c & d \\ a-c & b-d \end{vmatrix} \]

This simplifies to:
\[ = a(d - (b-d)) - b(c - (a-c)) + (cb - d(a-c)) \]
\[ = a(2d - b) - b(2c - a) + (cb - da + dc) \]

5. **Set the Determinant Equal to Zero**:
We need to set the entire expression equal to zero:
\[ a(2d - b) - b(2c - a) + (cb - da + dc) = 0 \]

6. **Simplify the Expression**:
After simplification, we can rearrange the terms to find a relationship:
\[ bc - ad = 0 \]
This implies:
\[ bc = ad \]

### Final Result:
The relationship that must hold true for the points to be collinear is:
\[ bc - ad = 0 \quad \text{or} \quad bc = ad \]