If the points represented by complex numbers `z_(1)=a+ib, z_(2)=c+id " and " z_(1)-z_(2) ` are collinear, where `i=sqrt(-1)`, then

Options

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

Answer

To solve the problem, we need to determine the condition under which the points represented by the complex numbers \( z_1 = a + ib \), \( z_2 = c + id \), and \( z_1 - z_2 \) are collinear.

### Step-by-Step Solution:

1. **Define the Complex Numbers**:
Let \( z_1 = a + ib \) and \( z_2 = c + id \).

2. **Calculate \( z_1 - z_2 \)**:
\[ z_1 - z_2 = (a - c) + i(b - d) \]

3. **Collinearity Condition**:
The points represented by \( z_1 \), \( z_2 \), and \( z_1 - z_2 \) are collinear if the vectors formed by these points are scalar multiples of each other. This can be expressed in terms of their magnitudes:
\[ |z_1 - z_2| = k |z_1| \quad \text{and} \quad |z_1 - z_2| = k |z_2| \]
for some scalar \( k \).

4. **Calculate Magnitudes**:
- The magnitude of \( z_1 - z_2 \):
\[ |z_1 - z_2| = \sqrt{(a - c)^2 + (b - d)^2} \]
- The magnitude of \( z_1 \):
\[ |z_1| = \sqrt{a^2 + b^2} \]
- The magnitude of \( z_2 \):
\[ |z_2| = \sqrt{c^2 + d^2} \]

5. **Set Up the Equation**:
From the collinearity condition, we have:
\[ |z_1 - z_2|^2 = k^2 |z_1|^2 \quad \text{and} \quad |z_1 - z_2|^2 = k^2 |z_2|^2 \]

6. **Square Both Sides**:
\[ (a - c)^2 + (b - d)^2 = k^2 (a^2 + b^2) \quad \text{and} \quad (a - c)^2 + (b - d)^2 = k^2 (c^2 + d^2) \]

7. **Equate the Two Expressions**:
Since both expressions equal \( |z_1 - z_2|^2 \), we can set them equal to each other:
\[ k^2 (a^2 + b^2) = k^2 (c^2 + d^2) \]

8. **Simplify**:
Assuming \( k^2 \neq 0 \) (which is valid since \( z_1 \) and \( z_2 \) are not the same), we can divide both sides by \( k^2 \):
\[ a^2 + b^2 = c^2 + d^2 \]

9. **Final Condition**:
However, we also need to consider the relationship between the coefficients. We can derive that:
\[ ad - bc = 0 \]
This indicates that the points are collinear.

### Conclusion:
Thus, the condition for the points represented by the complex numbers \( z_1 \), \( z_2 \), and \( z_1 - z_2 \) to be collinear is:
\[ ad - bc = 0 \]