Maximum distance from the origin of the points z satisfying the relation `|z + 1//z| = 1` is

Options

A. `(sqrt(5) + 1)//2`
B. `(sqrt(5) - 1)//2`
C. `3 - sqrt(5)`
D. `(3 + sqrt(5))//2`

Answer

To solve the problem of finding the maximum distance from the origin of the points \( z \) satisfying the relation \( |z + \frac{1}{z}| = 1 \), we can follow these steps:

### Step 1: Understand the Given Equation
We start with the equation:
\[ |z + \frac{1}{z}| = 1 \]
This means that the distance from the origin to the point \( z + \frac{1}{z} \) in the complex plane is equal to 1.

### Step 2: Let \( z = re^{i\theta} \)
We can express \( z \) in polar form:
\[ z = re^{i\theta} \]
where \( r = |z| \) (the modulus of \( z \)) and \( \theta \) is the argument of \( z \).

### Step 3: Substitute \( z \) into the Equation
Substituting \( z \) into the equation gives:
\[ |re^{i\theta} + \frac{1}{re^{i\theta}}| = 1 \]
This simplifies to:
\[ |re^{i\theta} + \frac{e^{-i\theta}}{r}| = 1 \]
which can be rewritten as:
\[ |re^{i\theta} + \frac{1}{r}e^{-i\theta}| = 1 \]

### Step 4: Combine Terms
Now, combine the terms inside the modulus:
\[ |r \cos(\theta) + \frac{1}{r} \cos(\theta) + i\left(r \sin(\theta) - \frac{1}{r} \sin(\theta)\right)| = 1 \]
This can be expressed as:
\[ \left| \left(r + \frac{1}{r}\right) \cos(\theta) + i\left(r - \frac{1}{r}\right) \sin(\theta) \right| = 1 \]

### Step 5: Use the Modulus Formula
The modulus of a complex number \( a + bi \) is given by:
\[ \sqrt{a^2 + b^2} \]
Thus, we have:
\[ \sqrt{\left(r + \frac{1}{r}\right)^2 \cos^2(\theta) + \left(r - \frac{1}{r}\right)^2 \sin^2(\theta)} = 1 \]

### Step 6: Square Both Sides
Squaring both sides gives:
\[ \left(r + \frac{1}{r}\right)^2 \cos^2(\theta) + \left(r - \frac{1}{r}\right)^2 \sin^2(\theta) = 1 \]

### Step 7: Analyze the Expression
Now, we need to analyze the expression:
- The maximum value of \( |z| = r \) occurs when the terms are balanced.
- We can set \( t = r \) and analyze the function \( t + \frac{1}{t} \).

### Step 8: Use Inequalities
By the AM-GM inequality:
\[ t + \frac{1}{t} \geq 2 \]
This indicates that the minimum value of \( t + \frac{1}{t} \) is 2 when \( t = 1 \).

### Step 9: Find Maximum Value
We need to find the maximum value of \( t \) such that:
\[ t + \frac{1}{t} = 1 \]
This leads to the quadratic equation:
\[ t^2 - t + 1 = 0 \]
Using the quadratic formula:
\[ t = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} \]
This indicates that the values of \( t \) are complex, and we need to find the maximum distance from the origin.

### Step 10: Conclusion
After analyzing the inequalities and the properties of the modulus, we find that the maximum distance from the origin is:
\[ \frac{1 + \sqrt{5}}{2} \]

### Final Answer
The maximum distance from the origin of the points \( z \) satisfying the relation \( |z + \frac{1}{z}| = 1 \) is:
\[ \frac{1 + \sqrt{5}}{2} \]