The least value of `|z-3-4i|^(2)+|z+2-7i|^(2)+|z-5+2i|^(2)` occurs when z=

Options

A. `1+3i`
B. `3+3i`
C. `3+4i`
D. None of these

Explanation

To find the least value of the expression \( |z - 3 - 4i|^2 + |z + 2 - 7i|^2 + |z - 5 + 2i|^2 \), we will follow these steps:

1. **Substitute \( z \)**: Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. Therefore, we can rewrite the expression as:
\[ |(x + yi) - (3 + 4i)|^2 + |(x + yi) - (-2 + 7i)|^2 + |(x + yi) - (5 - 2i)|^2 \]

2. **Calculate the Modulus**: We can express each term:
- \( |z - (3 + 4i)|^2 = |(x - 3) + (y - 4)i|^2 = (x - 3)^2 + (y - 4)^2 \)
- \( |z - (-2 + 7i)|^2 = |(x + 2) + (y - 7)i|^2 = (x + 2)^2 + (y - 7)^2 \)
- \( |z - (5 - 2i)|^2 = |(x - 5) + (y + 2)i|^2 = (x - 5)^2 + (y + 2)^2 \)

3. **Combine the Terms**: Now, we can write the total expression:
\[ (x - 3)^2 + (y - 4)^2 + (x + 2)^2 + (y - 7)^2 + (x - 5)^2 + (y + 2)^2 \]

4. **Expand Each Square**:
- \( (x - 3)^2 = x^2 - 6x + 9 \)
- \( (y - 4)^2 = y^2 - 8y + 16 \)
- \( (x + 2)^2 = x^2 + 4x + 4 \)
- \( (y - 7)^2 = y^2 - 14y + 49 \)
- \( (x - 5)^2 = x^2 - 10x + 25 \)
- \( (y + 2)^2 = y^2 + 4y + 4 \)

5. **Combine Like Terms**:
\[ = 3x^2 + 3y^2 - 12x - 18y + 103 \]

6. **Find the Minimum Value**: To minimize \( 3x^2 + 3y^2 - 12x - 18y + 103 \), we can complete the square for both \( x \) and \( y \).

- For \( x \):
\[ 3(x^2 - 4x) = 3((x - 2)^2 - 4) = 3(x - 2)^2 - 12 \]
- For \( y \):
\[ 3(y^2 - 6y) = 3((y - 3)^2 - 9) = 3(y - 3)^2 - 27 \]

Therefore, the expression becomes:
\[ 3(x - 2)^2 + 3(y - 3)^2 + 103 - 12 - 27 = 3(x - 2)^2 + 3(y - 3)^2 + 64 \]

7. **Determine the Minimum**: The minimum value occurs when \( (x - 2)^2 = 0 \) and \( (y - 3)^2 = 0 \), which gives \( x = 2 \) and \( y = 3 \).

8. **Final Result**: Thus, the least value occurs when:
\[ z = 2 + 3i \]