If m be the least value of `|z-3+4i|^2+|z-5-2i|^2, z in C` attained at `z=z_0` , then the ordered pair `(|z_0|,m)` is equal to :

Options

A. `(sqrt13,10)`
B. `(sqrt17,20)`
C. `(sqrt17,10)`
D. `(sqrt13,20)`

Answer

To solve the problem, we need to minimize the expression \( |z - (3 - 4i)|^2 + |z - (5 + 2i)|^2 \) for \( z \in \mathbb{C} \).

### Step-by-Step Solution

1. **Understand the Expression**:
The expression \( |z - (3 - 4i)|^2 + |z - (5 + 2i)|^2 \) represents the sum of the squares of the distances from the point \( z \) in the complex plane to the points \( (3, -4) \) and \( (5, 2) \).

2. **Identify the Points**:
Let \( z_1 = 3 - 4i \) and \( z_2 = 5 + 2i \). We can plot these points in the Argand plane:
- \( z_1 \) is at \( (3, -4) \)
- \( z_2 \) is at \( (5, 2) \)

3. **Find the Midpoint**:
To minimize the sum of the squares of the distances, we can find the midpoint \( z_0 \) of the line segment connecting \( z_1 \) and \( z_2 \):
\[ z_0 = \left( \frac{3 + 5}{2}, \frac{-4 + 2}{2} \right) = \left( \frac{8}{2}, \frac{-2}{2} \right) = (4, -1) \]
Thus, \( z_0 = 4 - i \).

4. **Calculate the Modulus of \( z_0 \)**:
The modulus \( |z_0| \) is given by:
\[ |z_0| = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \]

5. **Calculate the Minimum Value \( m \)**:
Now we need to calculate \( m \) by substituting \( z_0 \) back into the original expression:
\[ m = |z_0 - z_1|^2 + |z_0 - z_2|^2 \]
First, calculate \( |z_0 - z_1| \):
\[ z_0 - z_1 = (4 - i) - (3 - 4i) = 1 + 3i \]
Thus,
\[ |z_0 - z_1|^2 = |1 + 3i|^2 = 1^2 + 3^2 = 1 + 9 = 10 \]
Next, calculate \( |z_0 - z_2| \):
\[ z_0 - z_2 = (4 - i) - (5 + 2i) = -1 - 3i \]
Thus,
\[ |z_0 - z_2|^2 = |-1 - 3i|^2 = (-1)^2 + (-3)^2 = 1 + 9 = 10 \]
Therefore,
\[ m = 10 + 10 = 20 \]

6. **Final Ordered Pair**:
The ordered pair \( (|z_0|, m) \) is:
\[ \left( \sqrt{17}, 20 \right) \]

### Final Answer
The ordered pair \( (|z_0|, m) \) is \( \left( \sqrt{17}, 20 \right) \).