Let A,B and C be three sets of complex numbers as defined below: `{:(,A={z:Im(z) ge 1}),(,B={z:abs(z-2-i)=3}),(,C={z:Re(1-i)z)=3sqrt(2)"where" i=sqrt(-1)):}`
Let z be any point in `A cap B cap C`. Then, `abs(z+1-i)^(2)+abs(z-5-i)^(2)` lies between

Options

A. 25 and 29
B. 30 and 34
C. 35 and 39
D. 40 and 44

Answer

To solve the problem, we need to analyze the sets A, B, and C defined by the complex numbers and find the intersection of these sets. We will then calculate the expression \( |z + 1 - i|^2 + |z - 5 - i|^2 \).

### Step-by-Step Solution:

1. **Define the Sets**:
- Set \( A \): \( A = \{ z : \text{Im}(z) \geq 1 \} \) implies that the imaginary part of \( z \) is at least 1. This means \( y \geq 1 \) if we write \( z = x + iy \).
- Set \( B \): \( B = \{ z : |z - (2 + i)| = 3 \} \) describes a circle centered at \( (2, 1) \) with a radius of 3. In terms of \( x \) and \( y \), this can be expressed as:
\[ |(x + iy) - (2 + i)| = 3 \implies |(x - 2) + i(y - 1)| = 3 \]
This leads to the equation:
\[ (x - 2)^2 + (y - 1)^2 = 9 \]

- Set \( C \): \( C = \{ z : \text{Re}((1 - i)z) = 3\sqrt{2} \} \) can be rewritten as:
\[ \text{Re}((1 - i)(x + iy)) = 3\sqrt{2} \]
Expanding this gives:
\[ \text{Re}(x + iy - ix - y) = x + y = 3\sqrt{2} \]

2. **Find the Intersection \( A \cap B \cap C \)**:
- From set \( A \), we have \( y \geq 1 \).
- From set \( B \), we have the circle equation:
\[ (x - 2)^2 + (y - 1)^2 = 9 \]
- From set \( C \), we have the line equation:
\[ x + y = 3\sqrt{2} \]

3. **Substitute \( y \) from Set \( C \) into Set \( B \)**:
- From \( C \), we can express \( y \) as:
\[ y = 3\sqrt{2} - x \]
- Substitute this into the circle equation from \( B \):
\[ (x - 2)^2 + ((3\sqrt{2} - x) - 1)^2 = 9 \]
Simplifying gives:
\[ (x - 2)^2 + (3\sqrt{2} - x - 1)^2 = 9 \]
\[ (x - 2)^2 + (3\sqrt{2} - x - 1)^2 = 9 \]

4. **Solve the Equation**:
- Expand and simplify:
\[ (x - 2)^2 + (3\sqrt{2} - x - 1)^2 = 9 \]
This will yield a quadratic equation in \( x \).

5. **Calculate \( |z + 1 - i|^2 + |z - 5 - i|^2 \)**:
- Let \( z = x + iy \) where \( y = 3\sqrt{2} - x \).
- Calculate:
\[ |z + 1 - i|^2 = |(x + 1) + i(y - 1)|^2 = (x + 1)^2 + (y - 1)^2 \]
\[ |z - 5 - i|^2 = |(x - 5) + i(y - 1)|^2 = (x - 5)^2 + (y - 1)^2 \]
- Add these two results together.

6. **Determine the Range**:
- The final result will give a specific range for \( |z + 1 - i|^2 + |z - 5 - i|^2 \).