The complex number z satisfying `|z+1|=|z-1|` and arg `(z-1)/(z+1)=pi/4`, is

Options

A. `(sqrt(2)+1)+0i`
B. `0+(sqrt(2)+1)i`
C. `0+(sqrt(2)-1)i`
D. `(-sqrt(2)+1)+0i`

Answer

To solve the problem, we need to find the complex number \( z \) that satisfies the conditions \( |z + 1| = |z - 1| \) and \( \arg \left( \frac{z - 1}{z + 1} \right) = \frac{\pi}{4} \).

### Step 1: Set up the complex number
Assume \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Apply the modulus condition
The first condition is \( |z + 1| = |z - 1| \).

Substituting \( z \):
\[ |x + iy + 1| = |x + iy - 1| \]
This simplifies to:
\[ |x + 1 + iy| = |x - 1 + iy| \]

### Step 3: Write the modulus in terms of real and imaginary parts
The modulus can be expressed as:
\[ \sqrt{(x + 1)^2 + y^2} = \sqrt{(x - 1)^2 + y^2} \]

### Step 4: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ (x + 1)^2 + y^2 = (x - 1)^2 + y^2 \]

### Step 5: Simplify the equation
Cancelling \( y^2 \) from both sides:
\[ (x + 1)^2 = (x - 1)^2 \]
Expanding both sides:
\[ x^2 + 2x + 1 = x^2 - 2x + 1 \]
Cancelling \( x^2 + 1 \) from both sides:
\[ 2x = -2x \]
This simplifies to:
\[ 4x = 0 \implies x = 0 \]

### Step 6: Substitute \( x \) back into \( z \)
Since \( x = 0 \), we have:
\[ z = 0 + iy = iy \]

### Step 7: Use the argument condition
Now we apply the second condition:
\[ \arg \left( \frac{z - 1}{z + 1} \right) = \frac{\pi}{4} \]
Substituting \( z = iy \):
\[ \arg \left( \frac{iy - 1}{iy + 1} \right) = \frac{\pi}{4} \]

### Step 8: Simplify the fraction
This can be rewritten as:
\[ \frac{iy - 1}{iy + 1} = \frac{-1 + iy}{1 + iy} \]

### Step 9: Calculate the argument
The argument of a complex number \( \frac{a + bi}{c + di} \) is given by:
\[ \arg(a + bi) - \arg(c + di) \]
Calculating the arguments:
- For \( -1 + iy \):
\[ \arg(-1 + iy) = \tan^{-1} \left( \frac{y}{-1} \right) = \tan^{-1}(-y) \]
- For \( 1 + iy \):
\[ \arg(1 + iy) = \tan^{-1} \left( \frac{y}{1} \right) = \tan^{-1}(y) \]

### Step 10: Set up the equation
Thus, we have:
\[ \tan^{-1}(-y) - \tan^{-1}(y) = \frac{\pi}{4} \]

### Step 11: Use the tangent addition formula
Using the identity \( \tan^{-1}(-y) = -\tan^{-1}(y) \):
\[ -\tan^{-1}(y) - \tan^{-1}(y) = \frac{\pi}{4} \]
This simplifies to:
\[ -2\tan^{-1}(y) = \frac{\pi}{4} \implies \tan^{-1}(y) = -\frac{\pi}{8} \]

### Step 12: Solve for \( y \)
Taking the tangent of both sides:
\[ y = \tan\left(-\frac{\pi}{8}\right) = -\tan\left(\frac{\pi}{8}\right) \]
Using the known value:
\[ \tan\left(\frac{\pi}{8}\right) = \sqrt{2} - 1 \implies y = -(\sqrt{2} - 1) = 1 - \sqrt{2} \]

### Final Result
Thus, the complex number \( z \) is:
\[ z = 0 + i(1 - \sqrt{2}) = i(1 - \sqrt{2}) \]

### Summary
The complex number \( z \) satisfying the given conditions is:
\[ \boxed{i(1 - \sqrt{2})} \]