All complex numbers 'z' which satisfy the relation `|z-|z+1||=|z+|z-1||` on the complex plane lie on the

Options

A. `y=x`
B. `y=-x`
C. circle `x^(2)+y^(2)=1`
D. line `x=0` or on a line segment joining `(-1,0) to `(1,0)`

Explanation

To solve the equation \( |z - |z + 1|| = |z + |z - 1|| \), we will break it down step by step.

### Step 1: Define the complex number
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit.

### Step 2: Rewrite the equation
We need to express \( |z + 1| \) and \( |z - 1| \):
- \( |z + 1| = |(x + 1) + yi| = \sqrt{(x + 1)^2 + y^2} \)
- \( |z - 1| = |(x - 1) + yi| = \sqrt{(x - 1)^2 + y^2} \)

### Step 3: Substitute into the equation
Substituting these into the original equation gives:
\[ |z - |z + 1|| = |z + |z - 1|| \]
This translates to:
\[ |x + yi - \sqrt{(x + 1)^2 + y^2}| = |x + yi + \sqrt{(x - 1)^2 + y^2}| \]

### Step 4: Analyze the absolute values
The left side \( |z - |z + 1|| \) can be analyzed as:
\[ |x + yi - \sqrt{(x + 1)^2 + y^2}| \]
And the right side \( |z + |z - 1|| \) can be analyzed as:
\[ |x + yi + \sqrt{(x - 1)^2 + y^2}| \]

### Step 5: Set up the conditions
To satisfy the equation, we need to consider the geometric interpretation of the absolute values. The expression \( |z - a| \) represents the distance from the point \( z \) to the point \( a \) in the complex plane.

### Step 6: Geometric interpretation
From the equation, we can interpret that the distances from \( z \) to the points \( |z + 1| \) and \( |z - 1| \) must be equal. This implies that \( z \) lies on the perpendicular bisector of the segment joining the points \( -1 \) and \( 1 \) on the real axis.

### Step 7: Conclusion
The perpendicular bisector of the segment joining \( -1 \) and \( 1 \) is the imaginary axis, where \( x = 0 \).

Thus, all complex numbers \( z \) satisfying the given relation lie on the line \( x = 0 \) in the complex plane.

### Final Answer
The locus of all complex numbers \( z \) satisfying the relation is the imaginary axis.

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