If `A(z_(1))` and `B(z_(2))` are two fixed points in the Argand plane the locus of point P(z) satisfying `|z-z_(1)|+|z-z_(2)|=|z_(1)-z_(2)|`, is

Options

A. line passing through A and B
B. line segment joining A and B
C. an ellipse
D. a circle

Explanation

To solve the problem, we need to analyze the given equation involving complex numbers in the Argand plane. The equation is:

\[ |z - z_1| + |z - z_2| = |z_1 - z_2| \]

where \( A(z_1) \) and \( B(z_2) \) are two fixed points in the Argand plane, and \( P(z) \) is a point whose locus we need to determine.

### Step-by-Step Solution:

1. **Understanding the Terms**:
- \( |z - z_1| \) represents the distance from point \( P(z) \) to point \( A(z_1) \).
- \( |z - z_2| \) represents the distance from point \( P(z) \) to point \( B(z_2) \).
- \( |z_1 - z_2| \) represents the distance between points \( A(z_1) \) and \( B(z_2) \).

2. **Interpreting the Equation**:
- The equation \( |z - z_1| + |z - z_2| = |z_1 - z_2| \) states that the sum of the distances from point \( P(z) \) to points \( A(z_1) \) and \( B(z_2) \) is equal to the distance between \( A \) and \( B \).

3. **Geometric Interpretation**:
- According to the triangle inequality, for any point \( P \) not lying on the line segment \( AB \), the sum of the distances \( PA + PB \) would be greater than \( AB \). Thus, the only scenario where \( PA + PB = AB \) holds true is when point \( P \) lies on the line segment connecting points \( A \) and \( B \).

4. **Conclusion**:
- Therefore, the locus of point \( P(z) \) that satisfies the given condition is the line segment connecting points \( A(z_1) \) and \( B(z_2) \).

### Final Answer:
The locus of point \( P(z) \) is the line segment joining points \( A(z_1) \) and \( B(z_2) \).

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