Let z be a complex number (not lying on x-axis) of maximum modulus such that `|z+1/z|=1`. Then,

Options

A. Im(z)=0
B. Re(z)=0
C. amp(z)=`pi`
D. Re(z)=1

Answer

To solve the problem, we need to find the complex number \( z \) such that \( |z + \frac{1}{z}| = 1 \) and \( z \) does not lie on the x-axis.

Let's denote \( z \) as \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of the complex number.

### Step 1: Express the condition in terms of \( x \) and \( y \)

We start with the given condition:
\[ |z + \frac{1}{z}| = 1 \]

Substituting \( z = x + iy \):
\[ z + \frac{1}{z} = (x + iy) + \frac{1}{x + iy} \]

To simplify \( \frac{1}{z} \), we multiply the numerator and denominator by the conjugate:
\[ \frac{1}{z} = \frac{1}{x + iy} \cdot \frac{x - iy}{x - iy} = \frac{x - iy}{x^2 + y^2} \]

Thus, we have:
\[ z + \frac{1}{z} = (x + iy) + \frac{x - iy}{x^2 + y^2} \]
\[ = x + iy + \frac{x}{x^2 + y^2} - \frac{iy}{x^2 + y^2} \]
\[ = \left( x + \frac{x}{x^2 + y^2} \right) + i \left( y - \frac{y}{x^2 + y^2} \right) \]

### Step 2: Separate real and imaginary parts

Let:
\[ u = x + \frac{x}{x^2 + y^2}, \quad v = y - \frac{y}{x^2 + y^2} \]

The modulus condition becomes:
\[ |u + iv| = 1 \implies \sqrt{u^2 + v^2} = 1 \implies u^2 + v^2 = 1 \]

### Step 3: Substitute and simplify

Substituting \( u \) and \( v \):
\[ \left( x + \frac{x}{x^2 + y^2} \right)^2 + \left( y - \frac{y}{x^2 + y^2} \right)^2 = 1 \]

### Step 4: Analyze the conditions

1. **Case 1:** Assume \( y = 0 \) (this means \( z \) lies on the x-axis).
- This leads to \( |z| = 1 \) which contradicts the condition that \( z \) does not lie on the x-axis. Thus, this case is invalid.

2. **Case 2:** Assume \( x = 0 \) (this means \( z \) lies on the y-axis).
- Then \( z = iy \) and we have:
\[ |iy + \frac{1}{iy}| = |iy - \frac{i}{y}| = |i(y - \frac{1}{y})| = |y - \frac{1}{y}| \]
- Setting this equal to 1 gives:
\[ |y - \frac{1}{y}| = 1 \]

### Step 5: Solve the equation

This leads to two cases:
1. \( y - \frac{1}{y} = 1 \)
- Multiply through by \( y \):
\[ y^2 - y - 1 = 0 \]
- Solving this using the quadratic formula:
\[ y = \frac{1 \pm \sqrt{5}}{2} \]

2. \( y - \frac{1}{y} = -1 \)
- Similarly, we get:
\[ y^2 + y - 1 = 0 \]
- Solving this gives:
\[ y = \frac{-1 \pm \sqrt{5}}{2} \]

### Conclusion

The complex number \( z \) can be expressed as:
\[ z = i \left( \frac{1 + \sqrt{5}}{2} \right) \quad \text{or} \quad z = i \left( \frac{-1 + \sqrt{5}}{2} \right) \]