The complex number z =x+iy which satisfies the equation `| ( z - 3i )/( z + 3i ) | = 1,` then on

Options

A. the X-axis
B. the straight line y = 3
C. a circle passing through origin
D. None of the above

Answer

To solve the equation \( \left| \frac{z - 3i}{z + 3i} \right| = 1 \) for the complex number \( z = x + iy \), we can follow these steps:

### Step 1: Understand the Equation
The equation \( \left| \frac{z - 3i}{z + 3i} \right| = 1 \) implies that the magnitudes of the numerator and denominator are equal. This leads us to the conclusion that \( |z - 3i| = |z + 3i| \).

### Step 2: Write the Magnitudes
Substituting \( z = x + iy \):
- The numerator becomes \( z - 3i = x + i(y - 3) \).
- The denominator becomes \( z + 3i = x + i(y + 3) \).

Thus, we have:
\[ |z - 3i| = \sqrt{x^2 + (y - 3)^2} \]
\[ |z + 3i| = \sqrt{x^2 + (y + 3)^2} \]

### Step 3: Set the Magnitudes Equal
Now, we set the two magnitudes equal:
\[ \sqrt{x^2 + (y - 3)^2} = \sqrt{x^2 + (y + 3)^2} \]

### Step 4: Square Both Sides
To eliminate the square roots, we square both sides:
\[ x^2 + (y - 3)^2 = x^2 + (y + 3)^2 \]

### Step 5: Simplify the Equation
Cancel \( x^2 \) from both sides:
\[ (y - 3)^2 = (y + 3)^2 \]

Expanding both sides:
\[ y^2 - 6y + 9 = y^2 + 6y + 9 \]

### Step 6: Rearrange the Equation
Now, simplify the equation:
\[ -6y + 9 = 6y + 9 \]
Subtract \( 9 \) from both sides:
\[ -6y = 6y \]
Combine like terms:
\[ -12y = 0 \]

### Step 7: Solve for \( y \)
Dividing both sides by \(-12\):
\[ y = 0 \]

### Step 8: Conclusion
Since \( y = 0 \), the complex number \( z \) lies on the real axis. Therefore, \( z = x + 0i = x \) where \( x \) can be any real number.

### Final Answer
The solution is that the complex number \( z \) lies on the real axis, which can be represented as:
\[ z = x \quad \text{for any real } x. \]

---