Suppose z is a complex number satisfying z−(1/z)=4+4i . Find the length (or modulus) of z+(1/z) .

Given the complex number \( z \) satisfying the equation

\[
z - \frac{1}{z} = 4 + 4i,
\]

we need to find the modulus (or length) of \( z + \frac{1}{z} \).

### Step 1: Let \( z = a + bi \), where \( a \) and \( b \) are real numbers.

The reciprocal \( \frac{1}{z} \) can be expressed as:

\[
\frac{1}{z} = \frac{1}{a + bi} = \frac{a - bi}{a^2 + b^2}
\]

Thus, the equation \( z - \frac{1}{z} = 4 + 4i \) becomes:

\[
(a + bi) - \frac{a - bi}{a^2 + b^2} = 4 + 4i
\]

### Step 2: Separate the equation into real and imaginary parts.

Let's express the equation as:

\[
a + bi - \left(\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i\right) = 4 + 4i
\]

Equating real and imaginary parts:

Real part:

\[
a - \frac{a}{a^2 + b^2} = 4
\]

Imaginary part:

\[
b + \frac{b}{a^2 + b^2} = 4
\]

### Step 3: Simplify and solve for \( a \) and \( b \).

Real part equation:

\[
a\left(1 - \frac{1}{a^2 + b^2}\right) = 4
\]

This simplifies to:

\[
a\left(\frac{a^2 + b^2 - 1}{a^2 + b^2}\right) = 4
\]

Imaginary part equation:

\[
b\left(1 + \frac{1}{a^2 + b^2}\right) = 4
\]

This simplifies to:

\[
b\left(\frac{a^2 + b^2 + 1}{a^2 + b^2}\right) = 4
\]

### Step 4: Find \( z + \frac{1}{z} \).

Now, we calculate \( z + \frac{1}{z} \):

\[
z + \frac{1}{z} = (a + bi) + \left(\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i\right)
\]

\[
= \left(a + \frac{a}{a^2 + b^2}\right) + \left(b - \frac{b}{a^2 + b^2}\right)i
\]

### Step 5: Calculate the modulus of \( z + \frac{1}{z} \).

The modulus is given by:

\[
\left| z + \frac{1}{z} \right| = \sqrt{\left(a + \frac{a}{a^2 + b^2}\right)^2 + \left(b - \frac{b}{a^2 + b^2}\right)^2}
\]

We know from the given condition:

\[
\left| z - \frac{1}{z} \right|^2 = |4 + 4i|^2 = 4^2 + 4^2 = 32
\]

Thus, the modulus of \( z + \frac{1}{z} \) is also \( \sqrt{32} = 4\sqrt{2} \).

### Final Answer:

The modulus of \( z + \frac{1}{z} \) is \( 4\sqrt{2} \).

Would you like to see more details, or have any questions? Here are 5 related questions to consider:

1. What is the general method to find the modulus of a complex number?
2. How do you find the modulus of a complex expression like \( z - \frac{1}{z} \)?
3. Can you derive a general formula for \( |z + \frac{1}{z}| \) in terms of \( |z| \)?
4. What is the geometric interpretation of the modulus of a complex number?
5. How does the modulus change when a complex number is inverted (i.e., \( \frac{1}{z} \))?

**Tip:** When solving complex equations, separating real and imaginary parts is often key to finding solutions.

Learn how to find the modulus of z + (1/z) given the complex number z satisfies the equation z - (1/z) = 4 + 4i. This problem explores complex number properties, including representation, reciprocals, and modulus calculation. Suitable for high school students.

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