if ` z_(1)=3+i and z_(2) = 2-i, " then" |(z_(1) +z_(2)-1)/(z_(1) -z_(2)+i)|`is

Options

A. `sqrt8/5`
B. `sqrt8/5`
C. `8/5`
D. `8/sqrt5`

Answer

To solve the problem, we need to evaluate the expression:

\[ \left| \frac{z_1 + z_2 - 1}{z_1 - z_2 + i} \right| \]

where \( z_1 = 3 + i \) and \( z_2 = 2 - i \).

### Step 1: Calculate \( z_1 + z_2 - 1 \)

First, we find \( z_1 + z_2 \):

\[ z_1 + z_2 = (3 + i) + (2 - i) = 3 + 2 + i - i = 5 \]

Now, subtract 1 from this result:

\[ z_1 + z_2 - 1 = 5 - 1 = 4 \]

### Step 2: Calculate \( z_1 - z_2 + i \)

Next, we calculate \( z_1 - z_2 \):

\[ z_1 - z_2 = (3 + i) - (2 - i) = 3 - 2 + i + i = 1 + 2i \]

Now, add \( i \):

\[ z_1 - z_2 + i = (1 + 2i) + i = 1 + 3i \]

### Step 3: Find the modulus of the numerator and denominator

Now we need to find the modulus of the numerator and the denominator.

**Numerator:**

\[ |z_1 + z_2 - 1| = |4| = 4 \]

**Denominator:**

To find \( |z_1 - z_2 + i| \):

\[ |1 + 3i| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \]

### Step 4: Combine the results

Now we can substitute these values back into our original expression:

\[ \left| \frac{z_1 + z_2 - 1}{z_1 - z_2 + i} \right| = \frac{|4|}{|1 + 3i|} = \frac{4}{\sqrt{10}} \]

### Step 5: Simplify the expression

To simplify \( \frac{4}{\sqrt{10}} \), we can multiply the numerator and denominator by \( \sqrt{10} \):

\[ \frac{4}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{4\sqrt{10}}{10} = \frac{2\sqrt{10}}{5} \]

### Final Answer

Thus, the final answer is:

\[ \frac{2\sqrt{10}}{5} \]