The value of ` |1/(2+i) - 1/(2 -i)|` is

Options

A. `-2/5`
B. ` 4/25`
C. ` 2/5`
D. 0

Answer

To solve the problem \( | \frac{1}{2+i} - \frac{1}{2-i} | \), we will follow these steps:

### Step 1: Find a common denominator
We need to combine the two fractions. The common denominator will be the product of the two denominators:
\[ (2+i)(2-i) \]

### Step 2: Write the expression with the common denominator
The expression can be rewritten as:
\[ \frac{1}{2+i} - \frac{1}{2-i} = \frac{(2-i) - (2+i)}{(2+i)(2-i)} \]

### Step 3: Simplify the numerator
Now, simplify the numerator:
\[ (2-i) - (2+i) = 2 - i - 2 - i = -2i \]

### Step 4: Simplify the denominator
Next, simplify the denominator:
\[ (2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 \]

### Step 5: Combine the results
Now we can combine the results:
\[ \frac{-2i}{5} \]

### Step 6: Find the modulus
Now, we need to find the modulus of this complex number:
\[ \left| \frac{-2i}{5} \right| = \frac{|-2i|}{|5|} = \frac{2}{5} \]

### Final Answer
Thus, the value of \( | \frac{1}{2+i} - \frac{1}{2-i} | \) is:
\[ \frac{2}{5} \]

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