Range of Value for Modulus Expression of Complex Number

To find the range of the expression $|z + 1 - i|^2 + |z - 5 - i|^2$ for any point $z \in A \cap B \cap C$, we first analyze the given sets and the expression geometrically.

The sets are defined as:

1. $A = \{z : \operatorname{Im} z \ge 1\}$
2. $B = \{z : |z - (2 + i)| = 3\}$, which represents a circle in the complex plane centered at $z_0 = 2 + i$ with radius $R = 3$.
3. $C = \{z : \operatorname{Re}((1 - i)z) = \sqrt{2}\}$, which represents a straight line.

Now, consider the expression to be evaluated:
$E = |z - z_1|^2 + |z - z_2|^2$
where $z_1 = -1 + i$ and $z_2 = 5 + i$.

Let us determine the midpoint of the segment joining $z_1$ and $z_2$:
$z_{\text{mid}} = \frac{z_1 + z_2}{2} = \frac{(-1 + i) + (5 + i)}{2} = \frac{4 + 2i}{2} = 2 + i$

Notice that $z_{\text{mid}} = z_0$, which is precisely the center of the circle described by set $B$.

By Apollonius' Theorem (or the parallelogram law for complex numbers):
For any point $z$ and points $z_1, z_2$ with midpoint $z_0$:
$|z - z_1|^2 + |z - z_2|^2 = 2|z - z_0|^2 + 2|z_1 - z_0|^2$

Since $z \in B$, the distance of $z$ from the center $z_0 = 2 + i$ is equal to the radius:
$|z - z_0| = |z - (2 + i)| = 3$

Also, the distance between $z_1$ and $z_0$ is:
$|z_1 - z_0| = |(-1 + i) - (2 + i)| = |-3| = 3$

Substituting these values into the formula:
$E = 2(3)^2 + 2(3)^2 = 2(9) + 2(9) = 18 + 18 = 36$

Alternatively, setting $z = x + iy$:
$(x - 2)^2 + (y - 1)^2 = 9$
The expression is:

$$\begin{aligned} |z + 1 - i|^2 + |z - 5 - i|^2 &= (x + 1)^2 + (y - 1)^2 + (x - 5)^2 + (y - 1)^2 \\ &= \left[(x + 1)^2 + (x - 5)^2\right] + 2(y - 1)^2 \\ &= \left[2x^2 - 8x + 26\right] + 2\left[9 - (x - 2)^2\right] \\ &= 2x^2 - 8x + 26 + 18 - 2(x^2 - 4x + 4) \\ &= 44 - 8 = 36 \end{aligned}$$

Hence, the expression is identically equal to $36$ for any point on the circle $B$ (and thus for any point in $A \cap B \cap C$).

The value $36$ lies between $35$ and $39$.

Therefore, the correct option is C.