Let z be a complex number satisfying `1/2 le |z| le 4` , then sum of greatest and least values of `|z+1/z|` is :

Options

A. `65/4`
B. `65/16`
C. `17/4`
D. 17

Answer

To solve the problem, we need to find the sum of the greatest and least values of \( |z + \frac{1}{z}| \) given that \( \frac{1}{2} \leq |z| \leq 4 \).

### Step 1: Represent \( z \)
Let \( z = re^{i\theta} \), where \( r = |z| \) and \( \theta \) is the argument of \( z \). Since \( \frac{1}{2} \leq r \leq 4 \), we will analyze the expression \( |z + \frac{1}{z}| \).

### Step 2: Calculate \( |z + \frac{1}{z}| \)
We can express \( |z + \frac{1}{z}| \) as follows:
\[ |z + \frac{1}{z}| = |re^{i\theta} + \frac{1}{re^{i\theta}}| = |re^{i\theta} + \frac{1}{r} e^{-i\theta}| \]
This simplifies to:
\[ |z + \frac{1}{z}| = |r \cos \theta + \frac{1}{r} \cos \theta + i\left(r \sin \theta - \frac{1}{r} \sin \theta\right)| \]
\[ = |(r + \frac{1}{r}) \cos \theta + i(r - \frac{1}{r}) \sin \theta| \]

### Step 3: Use the modulus formula
Using the modulus formula, we have:
\[ |z + \frac{1}{z}| = \sqrt{((r + \frac{1}{r}) \cos \theta)^2 + ((r - \frac{1}{r}) \sin \theta)^2} \]
Expanding this gives:
\[ = \sqrt{(r^2 + 2 + \frac{1}{r^2}) \cos^2 \theta + (r^2 - 2 + \frac{1}{r^2}) \sin^2 \theta} \]
\[ = \sqrt{(r^2 + \frac{1}{r^2}) + 2\cos^2 \theta - 2\sin^2 \theta} \]
\[ = \sqrt{(r^2 + \frac{1}{r^2}) + 2\cos(2\theta)} \]

### Step 4: Analyze the expression
The term \( 2\cos(2\theta) \) varies between -2 and 2. Thus, we can write:
\[ |z + \frac{1}{z}| = \sqrt{(r^2 + \frac{1}{r^2}) + 2\cos(2\theta)} \]
The minimum value occurs when \( \cos(2\theta) = -1 \) and the maximum value occurs when \( \cos(2\theta) = 1 \).

### Step 5: Find minimum and maximum values
1. **Minimum Value**:
\[ |z + \frac{1}{z}|_{\text{min}} = \sqrt{(r^2 + \frac{1}{r^2}) - 2} = \sqrt{(r - \frac{1}{r})^2} \]
This equals \( |r - \frac{1}{r}| \).

2. **Maximum Value**:
\[ |z + \frac{1}{z}|_{\text{max}} = \sqrt{(r^2 + \frac{1}{r^2}) + 2} = \sqrt{(r + \frac{1}{r})^2} \]
This equals \( |r + \frac{1}{r}| \).

### Step 6: Evaluate for boundaries of \( r \)
- For \( r = \frac{1}{2} \):
- Minimum: \( |r - \frac{1}{r}| = |\frac{1}{2} - 2| = \frac{3}{2} \)
- Maximum: \( |r + \frac{1}{r}| = |\frac{1}{2} + 2| = \frac{5}{2} \)

- For \( r = 4 \):
- Minimum: \( |r - \frac{1}{r}| = |4 - \frac{1}{4}| = \frac{15}{4} \)
- Maximum: \( |r + \frac{1}{r}| = |4 + \frac{1}{4}| = \frac{17}{4} \)

### Step 7: Find the overall minimum and maximum values
- The overall minimum value of \( |z + \frac{1}{z}| \) is \( \frac{3}{2} \).
- The overall maximum value of \( |z + \frac{1}{z}| \) is \( \frac{17}{4} \).

### Step 8: Sum of greatest and least values
Finally, we sum the greatest and least values:
\[ \text{Sum} = \frac{3}{2} + \frac{17}{4} = \frac{6}{4} + \frac{17}{4} = \frac{23}{4} \]

Thus, the answer is:
\[ \frac{23}{4} \]