Let z be a complex number such that `| z + (1)/(z)| = 2 ` If |z| ` = r_(1) and r_(2) " for arg " z = (pi)/(4)` than
`| r_(1) - r_(2)|` =

Options

A. `(1)/(sqrt(2))`
B. 1
C. `sqrt(2)`
D. 2

Answer

To solve the problem, we need to find the values of \( r_1 \) and \( r_2 \) such that \( |z + \frac{1}{z}| = 2 \) given that \( |z| = r_1 \) and \( \arg(z) = \frac{\pi}{4} \).

### Step-by-step Solution:

1. **Express \( z \)**:
Since \( \arg(z) = \frac{\pi}{4} \), we can express \( z \) in polar form:
\[ z = r e^{i \frac{\pi}{4}} = r \left( \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right) \]

2. **Calculate \( \frac{1}{z} \)**:
The reciprocal of \( z \) is:
\[ \frac{1}{z} = \frac{1}{r e^{i \frac{\pi}{4}}} = \frac{1}{r} e^{-i \frac{\pi}{4}} = \frac{1}{r} \left( \frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \]

3. **Add \( z \) and \( \frac{1}{z} \)**:
Now we add \( z \) and \( \frac{1}{z} \):
\[ z + \frac{1}{z} = r \left( \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right) + \frac{1}{r} \left( \frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \]
Simplifying this, we get:
\[ z + \frac{1}{z} = \left( r + \frac{1}{r} \right) \frac{1}{\sqrt{2}} + i \left( r - \frac{1}{r} \right) \frac{1}{\sqrt{2}} \]

4. **Calculate the modulus**:
The modulus of \( z + \frac{1}{z} \) is given by:
\[ |z + \frac{1}{z}| = \sqrt{ \left( \left( r + \frac{1}{r} \right) \frac{1}{\sqrt{2}} \right)^2 + \left( \left( r - \frac{1}{r} \right) \frac{1}{\sqrt{2}} \right)^2 } \]
This simplifies to:
\[ |z + \frac{1}{z}| = \frac{1}{\sqrt{2}} \sqrt{ \left( r + \frac{1}{r} \right)^2 + \left( r - \frac{1}{r} \right)^2 } \]

5. **Simplify the expression**:
Expanding the squares:
\[ \left( r + \frac{1}{r} \right)^2 + \left( r - \frac{1}{r} \right)^2 = r^2 + 2 + \frac{1}{r^2} + r^2 - 2 + \frac{1}{r^2} = 2r^2 + 2\frac{1}{r^2} \]
Thus:
\[ |z + \frac{1}{z}| = \frac{1}{\sqrt{2}} \sqrt{2 \left( r^2 + \frac{1}{r^2} \right)} = \sqrt{r^2 + \frac{1}{r^2}} \]

6. **Set up the equation**:
Given \( |z + \frac{1}{z}| = 2 \), we have:
\[ \sqrt{r^2 + \frac{1}{r^2}} = 2 \]
Squaring both sides:
\[ r^2 + \frac{1}{r^2} = 4 \]

7. **Multiply through by \( r^2 \)**:
\[ r^4 - 4r^2 + 1 = 0 \]
Let \( x = r^2 \):
\[ x^2 - 4x + 1 = 0 \]

8. **Use the quadratic formula**:
\[ x = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3} \]
Therefore:
\[ r_1^2 = 2 + \sqrt{3}, \quad r_2^2 = 2 - \sqrt{3} \]

9. **Find \( |r_1 - r_2| \)**:
We need to find \( |r_1 - r_2| \):
\[ |r_1 - r_2| = \sqrt{2 + \sqrt{3}} - \sqrt{2 - \sqrt{3}} \]

10. **Calculate the final result**:
To find the numerical value:
\[ |r_1 - r_2| = \sqrt{2 + \sqrt{3}} - \sqrt{2 - \sqrt{3}} = \sqrt{2} \quad \text{(after simplification)} \]

### Final Answer:
\[ |r_1 - r_2| = \sqrt{2} \]