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. `sqrt(8/5)`
C. `8/5`
D. `8/sqrt5`

Answer

To solve the problem, we need to calculate the expression \(|(z_1 + z_2 - 1)/(z_1 - z_2 + i)|\) where \(z_1 = 3 + i\) and \(z_2 = 2 - i\).

### Step-by-Step Solution:

1. **Calculate \(z_1 + z_2 - 1\)**:
\[ z_1 + z_2 - 1 = (3 + i) + (2 - i) - 1 \]
Combine the real and imaginary parts:
\[ = (3 + 2 - 1) + (i - i) = 4 + 0i = 4 \]

2. **Calculate \(z_1 - z_2 + i\)**:
\[ z_1 - z_2 + i = (3 + i) - (2 - i) + i \]
Simplify:
\[ = (3 - 2) + (i + i + i) = 1 + 3i \]

3. **Find the modulus of the numerator**:
The modulus of a real number \(4\) is:
\[ |4| = 4 \]

4. **Find the modulus of the denominator \(1 + 3i\)**:
The modulus of a complex number \(a + bi\) is given by \(\sqrt{a^2 + b^2}\):
\[ |1 + 3i| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \]

5. **Calculate the modulus of the entire expression**:
Using the property of moduli:
\[ \left| \frac{z_1 + z_2 - 1}{z_1 - z_2 + i} \right| = \frac{|z_1 + z_2 - 1|}{|z_1 - z_2 + i|} = \frac{4}{\sqrt{10}} \]

6. **Simplify the expression**:
To express it in a more simplified form:
\[ \frac{4}{\sqrt{10}} = \frac{4 \sqrt{10}}{10} = \frac{2 \sqrt{10}}{5} \]

### Final Answer:
The value of \(|(z_1 + z_2 - 1)/(z_1 - z_2 + i)|\) is \(\frac{2 \sqrt{10}}{5}\).