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 given problem, we need to find the maximum value of the modulus of the complex number \( z \) that satisfies the equation:

\[ |z|^2 + \frac{4}{|z|^2} - 2\left(\frac{z}{\bar{z}} + \frac{\bar{z}}{z}\right) - 16 = 0 \]

Let's denote \( |z| = r \). We can express \( z \) in polar form as \( z = r(\cos \theta + i \sin \theta) \), where \( \bar{z} = r(\cos \theta - i \sin \theta) \).

### Step 1: Rewrite the equation

First, we rewrite the term \( \frac{z}{\bar{z}} + \frac{\bar{z}}{z} \):

\[ \frac{z}{\bar{z}} = \frac{r(\cos \theta + i \sin \theta)}{r(\cos \theta - i \sin \theta)} = \frac{\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta} \]

This simplifies to:

\[ \frac{z}{\bar{z}} + \frac{\bar{z}}{z} = 2 \cos(2\theta) \]

### Step 2: Substitute into the equation

Now substitute this back into the original equation:

\[ |z|^2 + \frac{4}{|z|^2} - 2(2 \cos(2\theta)) - 16 = 0 \]

This simplifies to:

\[ r^2 + \frac{4}{r^2} - 4 \cos(2\theta) - 16 = 0 \]

### Step 3: Rearranging the equation

Rearranging gives:

\[ r^2 + \frac{4}{r^2} - 4 \cos(2\theta) = 16 \]

### Step 4: Multiply through by \( r^2 \)

To eliminate the fraction, multiply through by \( r^2 \):

\[ r^4 - 16r^2 + 4 - 4r^2 \cos(2\theta) = 0 \]

### Step 5: Factor the equation

Rearranging gives:

\[ r^4 - (16 + 4 \cos(2\theta))r^2 + 4 = 0 \]

Let \( x = r^2 \), then we have a quadratic equation:

\[ x^2 - (16 + 4 \cos(2\theta))x + 4 = 0 \]

### Step 6: Use the quadratic formula

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ x = \frac{(16 + 4 \cos(2\theta)) \pm \sqrt{(16 + 4 \cos(2\theta))^2 - 16}}{2} \]

### Step 7: Find the maximum value of \( r^2 \)

To maximize \( r^2 \), we need to maximize \( \cos(2\theta) \). The maximum value of \( \cos(2\theta) \) is 1.

Substituting \( \cos(2\theta) = 1 \):

\[ x = \frac{(16 + 4 \cdot 1) \pm \sqrt{(16 + 4)^2 - 16}}{2} \]
\[ x = \frac{20 \pm \sqrt{400 - 16}}{2} \]
\[ x = \frac{20 \pm \sqrt{384}}{2} \]
\[ x = \frac{20 \pm 8\sqrt{6}}{2} \]
\[ x = 10 \pm 4\sqrt{6} \]

### Step 8: Choose the maximum value

The maximum value of \( r^2 \) is \( 10 + 4\sqrt{6} \).

### Step 9: Find \( r \)

Thus, the maximum value of \( |z| \) is:

\[ |z| = \sqrt{10 + 4\sqrt{6}} \]

### Final Answer

The maximum value of \( |z| \) is \( \sqrt{10 + 4\sqrt{6}} \).

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