If a complex number `z` satisfies `|z|^(2)+(4)/(|z|)^(2)-2((z)/(barz)+(barz)/(z))-16=0`, then the maximum value of `|z|` is

Options

A. `sqrt(6)+1`
B. `4`
C. `2+sqrt(6)`
D. `6`

Explanation

To solve the equation given in the problem, we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ |z|^2 + \frac{4}{|z|^2} - 2\left(\frac{z}{\bar{z}} + \frac{\bar{z}}{z}\right) - 16 = 0 \]
Let \( r = |z| \). Then, we can express the equation in terms of \( r \).

### Step 2: Simplify the terms
We know that:
\[ \frac{z}{\bar{z}} = \frac{z}{|z|^2} \cdot |z|^2 = e^{i\theta} \quad \text{(where \( z = re^{i\theta} \))} \]
Thus, we have:
\[ \frac{z}{\bar{z}} + \frac{\bar{z}}{z} = e^{i\theta} + e^{-i\theta} = 2\cos(\theta) \]
Substituting this back into the equation gives:
\[ r^2 + \frac{4}{r^2} - 4\cos(\theta) - 16 = 0 \]

### Step 3: Multiply through by \( r^2 \)
To eliminate the fraction, we multiply the entire equation by \( r^2 \):
\[ r^4 + 4 - 4r^2\cos(\theta) - 16r^2 = 0 \]
Rearranging gives:
\[ r^4 - 4r^2\cos(\theta) - 16r^2 + 4 = 0 \]

### Step 4: Rearranging the quadratic equation
We can factor out \( r^2 \):
\[ r^4 - (4\cos(\theta) + 16)r^2 + 4 = 0 \]
Let \( x = r^2 \). The equation becomes:
\[ x^2 - (4\cos(\theta) + 16)x + 4 = 0 \]

### Step 5: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{4\cos(\theta) + 16 \pm \sqrt{(4\cos(\theta) + 16)^2 - 16}}{2} \]

### Step 6: Simplifying the discriminant
Calculating the discriminant:
\[ (4\cos(\theta) + 16)^2 - 16 = 16\cos^2(\theta) + 128\cos(\theta) + 256 - 16 = 16\cos^2(\theta) + 128\cos(\theta) + 240 \]

### Step 7: Maximizing \( r^2 \)
To find the maximum value of \( r \), we need to maximize \( \cos(\theta) \). The maximum value of \( \cos(\theta) \) is 1. Substituting \( \cos(\theta) = 1 \):
\[ x = \frac{4(1) + 16 \pm \sqrt{(4(1) + 16)^2 - 16}}{2} \]
This simplifies to:
\[ x = \frac{20 \pm \sqrt{400 - 16}}{2} = \frac{20 \pm \sqrt{384}}{2} \]
Calculating \( \sqrt{384} = 8\sqrt{6} \):
\[ x = \frac{20 \pm 8\sqrt{6}}{2} = 10 \pm 4\sqrt{6} \]

### Step 8: Finding \( r \)
Since \( r^2 = 10 + 4\sqrt{6} \) gives the maximum value:
\[ r = \sqrt{10 + 4\sqrt{6}} \]

### Step 9: Final simplification
We can express \( r \) as:
\[ r = 2 + \sqrt{6} \]

### Conclusion
Thus, the maximum value of \( |z| \) is:
\[ \boxed{2 + \sqrt{6}} \]