If conjugate of a complex number z is `(2+5i)/(4-3i)`, then |Re(z) + Im(z)| is equal to ____________

Answer

To solve the problem, we need to find the value of \(|\text{Re}(z) + \text{Im}(z)|\) given that the conjugate of the complex number \( z \) is \(\frac{2 + 5i}{4 - 3i}\).

### Step-by-Step Solution:

1. **Identify the conjugate of z**:
We know that the conjugate of \( z \) is given as:
\[ z^* = \frac{2 + 5i}{4 - 3i} \]

2. **Multiply by the conjugate of the denominator**:
To simplify this expression, we multiply the numerator and the denominator by the conjugate of the denominator:
\[ z^* = \frac{(2 + 5i)(4 + 3i)}{(4 - 3i)(4 + 3i)} \]

3. **Calculate the denominator**:
The denominator can be calculated as follows:
\[ (4 - 3i)(4 + 3i) = 4^2 - (3i)^2 = 16 - (-9) = 16 + 9 = 25 \]

4. **Calculate the numerator**:
Now, we calculate the numerator:
\[ (2 + 5i)(4 + 3i) = 2 \cdot 4 + 2 \cdot 3i + 5i \cdot 4 + 5i \cdot 3i \]
\[ = 8 + 6i + 20i + 15(-1) = 8 + 26i - 15 = -7 + 26i \]

5. **Combine the results**:
Now we can write \( z^* \):
\[ z^* = \frac{-7 + 26i}{25} = -\frac{7}{25} + \frac{26}{25}i \]

6. **Find z**:
The complex number \( z \) is the conjugate of \( z^* \):
\[ z = -\frac{7}{25} - \frac{26}{25}i \]

7. **Extract the real and imaginary parts**:
From \( z = -\frac{7}{25} - \frac{26}{25}i \), we have:
- Real part, \(\text{Re}(z) = -\frac{7}{25}\)
- Imaginary part, \(\text{Im}(z) = -\frac{26}{25}\)

8. **Calculate \(|\text{Re}(z) + \text{Im}(z)|\)**:
Now, we calculate:
\[ \text{Re}(z) + \text{Im}(z) = -\frac{7}{25} - \frac{26}{25} = -\frac{33}{25} \]
Taking the modulus:
\[ |\text{Re}(z) + \text{Im}(z)| = \left| -\frac{33}{25} \right| = \frac{33}{25} \]

### Final Answer:
\[ |\text{Re}(z) + \text{Im}(z)| = \frac{33}{25} \]