If the ratio `(1-z)/(1+z)` is pyrely imaginary, then

Options

A. `0 lt |z|lt 1`
B. `|z|=1`
C. `|z|gt 1`
D. bounds for |z| can not be decided

Answer

To solve the problem, we need to determine the conditions under which the expression \(\frac{1-z}{1+z}\) is purely imaginary.

### Step-by-Step Solution:

1. **Substitute \( z \) with \( x + iy \)**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then we can rewrite the expression:
\[ \frac{1 - z}{1 + z} = \frac{1 - (x + iy)}{1 + (x + iy)} = \frac{1 - x - iy}{1 + x + iy} \]

2. **Multiply by the Conjugate**:
To simplify this expression, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(1 - x - iy)(1 + x - iy)}{(1 + x + iy)(1 + x - iy)} \]

3. **Simplify the Denominator**:
The denominator simplifies as follows:
\[ (1 + x)^2 + y^2 = (1 + x)^2 + y^2 \]

4. **Simplify the Numerator**:
The numerator simplifies as follows:
\[ (1 - x)(1 + x) - iy(1 + x) + iy(1 - x) = (1 - x^2) + i(-y(1 + x) + y(1 - x)) \]
This simplifies to:
\[ (1 - x^2) + i(-2xy) \]

5. **Combine the Results**:
Now we have:
\[ \frac{(1 - x^2) + i(-2y)}{(1 + x)^2 + y^2} \]

6. **Determine the Real Part**:
For the expression to be purely imaginary, the real part must be zero:
\[ 1 - x^2 = 0 \]
This implies:
\[ x^2 = 1 \quad \Rightarrow \quad x = \pm 1 \]

7. **Determine the Imaginary Part**:
The imaginary part does not need to be zero, but we need to ensure that the overall expression remains purely imaginary. Thus, we can focus on the condition derived from the real part.

8. **Using the Magnitude**:
Since \( x^2 + y^2 = 1 \) (from the earlier steps), we can conclude that:
\[ |z|^2 = x^2 + y^2 = 1 \quad \Rightarrow \quad |z| = 1 \]

### Conclusion:
The condition for the ratio \(\frac{1-z}{1+z}\) to be purely imaginary is that the modulus of \( z \) is equal to 1.

### Final Answer:
\[ \text{The answer is } |z| = 1. \]