Find the complex number z with maximum and minimum possible values of `|z|` satisfying
(a) `|z + (1)/(z) | =1`.

Answer

To find the complex number \( z \) with maximum and minimum possible values of \( |z| \) satisfying the equation

\[ |z + \frac{1}{z}| = 1, \]

we will follow these steps:

### Step 1: Rewrite the equation
Let \( z = re^{i\theta} \), where \( r = |z| \). Then, we can express \( \frac{1}{z} \) as \( \frac{1}{re^{i\theta}} = \frac{1}{r} e^{-i\theta} \). The equation becomes:

\[ |re^{i\theta} + \frac{1}{r} e^{-i\theta}| = 1. \]

### Step 2: Simplify the expression
Now, we can rewrite the left-hand side:

\[ |re^{i\theta} + \frac{1}{r} e^{-i\theta}| = |r e^{i\theta} + \frac{1}{r} e^{-i\theta}|. \]

Using the property of modulus, we have:

\[ |z + \frac{1}{z}| = |r e^{i\theta} + \frac{1}{r} e^{-i\theta}| = |r + \frac{1}{r} e^{-2i\theta}|. \]

### Step 3: Apply the triangle inequality
Using the triangle inequality, we know:

\[ |z + \frac{1}{z}| \leq |z| + |\frac{1}{z}| = r + \frac{1}{r}. \]

From the equation \( |z + \frac{1}{z}| = 1 \), we can write:

\[ 1 \leq r + \frac{1}{r}. \]

### Step 4: Solve the inequality
The inequality \( r + \frac{1}{r} \geq 1 \) holds for \( r > 0 \). To find the minimum and maximum values of \( r \), we can rearrange this to:

\[ r^2 - r + 1 \geq 0. \]

### Step 5: Use the quadratic formula
The roots of the equation \( r^2 - r - 1 = 0 \) can be found using the quadratic formula:

\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{1 \pm \sqrt{5}}{2}. \]

### Step 6: Determine the maximum and minimum values
Thus, we have two values:

1. \( r_{\text{min}} = \frac{1 - \sqrt{5}}{2} \)
2. \( r_{\text{max}} = \frac{1 + \sqrt{5}}{2} \)

### Step 7: Conclusion
Since \( r \) must be positive, we discard \( r_{\text{min}} \) as it is negative. Therefore, the minimum value of \( |z| \) is:

\[ |z|_{\text{min}} = \frac{1 - \sqrt{5}}{2} \quad \text{(not valid, so we take the next valid value)} \]

The maximum value of \( |z| \) is:

\[ |z|_{\text{max}} = \frac{1 + \sqrt{5}}{2}. \]

### Final Result
Thus, the maximum and minimum possible values of \( |z| \) satisfying the condition are:

- Minimum: \( |z|_{\text{min}} = 1 - \frac{\sqrt{5}}{2} \) (not valid, take \( 0 \))
- Maximum: \( |z|_{\text{max}} = \frac{1 + \sqrt{5}}{2} \).