If `z_(1) , z_(2), z_(3)` are three complex numbers in A.P., then they lie on :

Options

A. a circle
B. a straight line
C. a parabola
D. are ellipse

Answer

To solve the problem, we need to show that if three complex numbers \( z_1, z_2, z_3 \) are in arithmetic progression (A.P.), then they lie on a straight line in the complex plane.

### Step-by-Step Solution:

1. **Understanding A.P.**:
By definition, three numbers \( z_1, z_2, z_3 \) are in A.P. if:
\[ z_2 - z_1 = z_3 - z_2 \]

2. **Rearranging the A.P. Condition**:
From the above equation, we can rearrange it to:
\[ z_2 - z_1 = z_3 - z_2 \implies z_2 - z_1 = -(z_2 - z_3) \]

3. **Expressing the Relationship**:
This can be rewritten as:
\[ z_2 - z_1 = -1 \cdot (z_2 - z_3) \]

4. **Taking Arguments**:
Now, we take the argument of both sides:
\[ \text{arg}(z_2 - z_1) = \text{arg}(-1) + \text{arg}(z_2 - z_3) \]
Since the argument of \(-1\) is \( \pi \) (or \( 180^\circ \)), we have:
\[ \text{arg}(z_2 - z_1) = \pi + \text{arg}(z_2 - z_3) \]

5. **Understanding the Implication**:
The equation \( \text{arg}(z_2 - z_1) = \pi + \text{arg}(z_2 - z_3) \) implies that the vectors \( z_2 - z_1 \) and \( z_2 - z_3 \) are in opposite directions. This means that the points \( z_1, z_2, z_3 \) are collinear.

6. **Conclusion**:
Therefore, if \( z_1, z_2, z_3 \) are in A.P., they lie on a straight line in the complex plane.

### Final Answer:
The three complex numbers \( z_1, z_2, z_3 \) lie on a straight line.

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