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 \( |z - (3 + 4i)|^2 + |z + (2 - 7i)|^2 + |z - (5 - 2i)|^2 \), we can follow these steps:

### Step 1: Define \( z \)
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers.

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ |z - (3 + 4i)|^2 + |z + (2 - 7i)|^2 + |z - (5 - 2i)|^2 \]
This becomes:
\[ | (x - 3) + (y - 4)i |^2 + | (x + 2) + (y + 7)i |^2 + | (x - 5) + (y + 2)i |^2 \]

### Step 3: Expand each modulus
Now, we expand each modulus squared:
1. \( |z - (3 + 4i)|^2 = (x - 3)^2 + (y - 4)^2 \)
2. \( |z + (2 - 7i)|^2 = (x + 2)^2 + (y + 7)^2 \)
3. \( |z - (5 - 2i)|^2 = (x - 5)^2 + (y + 2)^2 \)

### Step 4: Combine the expressions
Combine these three expressions:
\[ (x - 3)^2 + (y - 4)^2 + (x + 2)^2 + (y + 7)^2 + (x - 5)^2 + (y + 2)^2 \]

### Step 5: Simplify the combined expression
Now, we simplify:
\[ = (x^2 - 6x + 9 + y^2 - 8y + 16) + (x^2 + 4x + 4 + y^2 + 14y + 49) + (x^2 - 10x + 25 + y^2 + 4y + 4) \]
Combine like terms:
\[ = 3x^2 + 3y^2 - 12x + 10y + 78 \]

### Step 6: Complete the square
Now, we complete the square for \( x \) and \( y \):
1. For \( x \):
\[ 3(x^2 - 4x) = 3((x - 2)^2 - 4) = 3(x - 2)^2 - 12 \]
2. For \( y \):
\[ 3(y^2 + \frac{10}{3}y) = 3\left((y + \frac{5}{3})^2 - \frac{25}{9}\right) = 3(y + \frac{5}{3})^2 - \frac{25}{3} \]

### Step 7: Substitute back
Substituting back into the expression gives:
\[ = 3(x - 2)^2 + 3\left(y + \frac{5}{3}\right)^2 + 78 - 12 - \frac{25}{3} \]
\[ = 3(x - 2)^2 + 3\left(y + \frac{5}{3}\right)^2 + \frac{234 - 36 - 25}{3} \]
\[ = 3(x - 2)^2 + 3\left(y + \frac{5}{3}\right)^2 + \frac{173}{3} \]

### Step 8: Find the minimum value
The minimum value occurs when \( (x - 2)^2 = 0 \) and \( \left(y + \frac{5}{3}\right)^2 = 0 \), which gives:
\[ x = 2, \quad y = -\frac{5}{3} \]
Thus, \( z = 2 - \frac{5}{3}i \).

### Final Answer
The least value of the expression occurs when:
\[ z = 2 - \frac{5}{3}i \]