The complex number z which satisfies the condition `|(i +z)/(i-z)|=1` lies on:

Options

A. circle ` x^(2) + y^(2) = 1`
B. the X-axis
C. the Y-axis
D. the line x + y = 1

Answer

To solve the problem, we need to find the complex number \( z \) that satisfies the condition \( \left| \frac{i + z}{i - z} \right| = 1 \).

### Step-by-Step Solution:

1. **Represent the Complex Number**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

2. **Substitute \( z \) into the Expression**:
Substitute \( z \) into the expression:
\[ \frac{i + z}{i - z} = \frac{i + (x + iy)}{i - (x + iy)} = \frac{i + x + iy}{i - x - iy} \]
This simplifies to:
\[ \frac{(x + iy + i)}{(-x + i(1 - y))} \]
Which can be rewritten as:
\[ \frac{x + (y + 1)i}{-x + (1 - y)i} \]

3. **Calculate the Modulus**:
The modulus of a complex number \( \frac{a + bi}{c + di} \) can be expressed as:
\[ \left| \frac{a + bi}{c + di} \right| = \frac{\sqrt{a^2 + b^2}}{\sqrt{c^2 + d^2}} \]
Here, \( a = x \), \( b = y + 1 \), \( c = -x \), and \( d = 1 - y \). Thus,
\[ \left| \frac{i + z}{i - z} \right| = \frac{\sqrt{x^2 + (y + 1)^2}}{\sqrt{x^2 + (1 - y)^2}} \]

4. **Set the Modulus Equal to 1**:
Since we know \( \left| \frac{i + z}{i - z} \right| = 1 \), we can set up the equation:
\[ \frac{\sqrt{x^2 + (y + 1)^2}}{\sqrt{x^2 + (1 - y)^2}} = 1 \]

5. **Square Both Sides**:
Squaring both sides gives:
\[ x^2 + (y + 1)^2 = x^2 + (1 - y)^2 \]

6. **Simplify the Equation**:
Cancel \( x^2 \) from both sides:
\[ (y + 1)^2 = (1 - y)^2 \]
Expanding both sides:
\[ y^2 + 2y + 1 = 1 - 2y + y^2 \]

7. **Solve for \( y \)**:
Cancel \( y^2 \) from both sides:
\[ 2y + 1 = 1 - 2y \]
Rearranging gives:
\[ 4y = 0 \implies y = 0 \]

8. **Conclusion**:
Since \( y = 0 \), the complex number \( z \) lies on the real axis, which is represented by the equation \( y = 0 \) (the x-axis).

### Final Answer:
The complex number \( z \) lies on the x-axis.