If `| z - ( 4)/( z)| =2`, then the maximum value o f `|z|` is equal to `:`

Options

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

Answer

To solve the equation \( | z - \frac{4}{z} | = 2 \) and find the maximum value of \( |z| \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ | z - \frac{4}{z} | = 2 \]
Let \( |z| = r \). Then, we can express \( z \) in terms of \( r \):
\[ |z| = r \implies |z - \frac{4}{z}| = |z| - |\frac{4}{z}| = r - \frac{4}{r} \]
Thus, the equation becomes:
\[ |r - \frac{4}{r}| = 2 \]

### Step 2: Set up inequalities
This absolute value equation gives us two cases:
1. \( r - \frac{4}{r} = 2 \)
2. \( r - \frac{4}{r} = -2 \)

### Step 3: Solve the first case
**Case 1:**
\[ r - \frac{4}{r} = 2 \]
Multiplying through by \( r \) (assuming \( r > 0 \)):
\[ r^2 - 4 = 2r \implies r^2 - 2r - 4 = 0 \]
Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ r = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 + 16}}{2} = \frac{2 \pm \sqrt{20}}{2} \]
\[ = 1 \pm \sqrt{5} \]
Thus, we have two potential values for \( r \):
\[ r_1 = 1 + \sqrt{5}, \quad r_2 = 1 - \sqrt{5} \quad (\text{discarded since } r > 0) \]

### Step 4: Solve the second case
**Case 2:**
\[ r - \frac{4}{r} = -2 \]
Again, multiplying through by \( r \):
\[ r^2 - 4 = -2r \implies r^2 + 2r - 4 = 0 \]
Using the quadratic formula:
\[ r = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 + 16}}{2} = \frac{-2 \pm \sqrt{20}}{2} \]
\[ = -1 \pm \sqrt{5} \]
Thus, we have two potential values for \( r \):
\[ r_3 = -1 + \sqrt{5}, \quad r_4 = -1 - \sqrt{5} \quad (\text{discarded since } r > 0) \]

### Step 5: Determine maximum value of \( |z| \)
Now we have two valid values for \( r \):
1. From Case 1: \( r = 1 + \sqrt{5} \)
2. From Case 2: \( r = -1 + \sqrt{5} \)

To find the maximum value of \( |z| \):
\[ \text{Maximum } |z| = \max(1 + \sqrt{5}, -1 + \sqrt{5}) \]
Since \( 1 + \sqrt{5} > -1 + \sqrt{5} \), the maximum value is:
\[ |z|_{\text{max}} = 1 + \sqrt{5} \]

### Final Answer
The maximum value of \( |z| \) is \( 1 + \sqrt{5} \).

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