If `|z|= "min" {|z-1|, |z+1|}`, then

Options

A. `|z+ bar(z)|=(1)/(2)`
B. `z + bar(z) =1`
C. `|z + bar(z) | =1`
D. None of these

Answer

To solve the problem where \( |z| = \min \{ |z - 1|, |z + 1| \} \), we can break it down step by step.

### Step 1: Understand the problem
We are given that the modulus of \( z \) is equal to the minimum of the distances from \( z \) to the points \( 1 \) and \( -1 \) in the complex plane. This means we need to find the points \( z \) such that the distance to the origin (which is \( |z| \)) is minimized compared to the distances to \( 1 \) and \( -1 \).

### Step 2: Set up the equations
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The distances can be expressed as:
- \( |z - 1| = |(x - 1) + iy| = \sqrt{(x - 1)^2 + y^2} \)
- \( |z + 1| = |(x + 1) + iy| = \sqrt{(x + 1)^2 + y^2} \)
- \( |z| = \sqrt{x^2 + y^2} \)

### Step 3: Set up the inequalities
We need to satisfy the condition:
\[ |z| = \min \{ |z - 1|, |z + 1| \} \]
This leads us to two cases to consider:
1. \( |z| = |z - 1| \)
2. \( |z| = |z + 1| \)

### Step 4: Solve the first case \( |z| = |z - 1| \)
From \( |z| = |z - 1| \):
\[ \sqrt{x^2 + y^2} = \sqrt{(x - 1)^2 + y^2} \]
Squaring both sides gives:
\[ x^2 + y^2 = (x - 1)^2 + y^2 \]
This simplifies to:
\[ x^2 + y^2 = x^2 - 2x + 1 + y^2 \]
Cancelling \( x^2 \) and \( y^2 \) from both sides, we get:
\[ 0 = -2x + 1 \implies x = \frac{1}{2} \]

### Step 5: Solve the second case \( |z| = |z + 1| \)
From \( |z| = |z + 1| \):
\[ \sqrt{x^2 + y^2} = \sqrt{(x + 1)^2 + y^2} \]
Squaring both sides gives:
\[ x^2 + y^2 = (x + 1)^2 + y^2 \]
This simplifies to:
\[ x^2 + y^2 = x^2 + 2x + 1 + y^2 \]
Cancelling \( x^2 \) and \( y^2 \) from both sides, we get:
\[ 0 = 2x + 1 \implies x = -\frac{1}{2} \]

### Step 6: Conclusion
We have two possible values for \( x \):
1. \( x = \frac{1}{2} \)
2. \( x = -\frac{1}{2} \)

Since \( |z| \) is the distance from the origin, we can conclude that \( |z| \) must be equal to the minimum distance from the origin to the line segment joining \( (1, 0) \) and \( (-1, 0) \). The minimum distance occurs at the midpoint of this segment, which is \( 0 \).

Thus, the final answer is:
\[ |z| = 1 \quad \text{(the distance from the origin to the points 1 and -1)} \]