If z be a complex number, then
`|z-3-4i|^(2)+|z+4+2i|^(2)=k` represents a circle, if k is equal to

Options

A. 30
B. 40
C. 55
D. 35

Answer

To solve the problem, we need to analyze the expression given and determine the value of \( k \) such that the equation represents a circle.

### Step-by-Step Solution:

1. **Let \( z = x + iy \)**:
We start by substituting \( z \) with its real and imaginary parts, where \( x \) is the real part and \( y \) is the imaginary part.

\[ |z - 3 - 4i|^2 + |z + 4 + 2i|^2 = k \]

2. **Rewrite the modulus expressions**:
We can express the moduli as follows:
\[ |z - 3 - 4i|^2 = |(x - 3) + i(y - 4)|^2 = (x - 3)^2 + (y - 4)^2 \]
\[ |z + 4 + 2i|^2 = |(x + 4) + i(y + 2)|^2 = (x + 4)^2 + (y + 2)^2 \]

3. **Combine the expressions**:
Substitute these back into the equation:
\[ (x - 3)^2 + (y - 4)^2 + (x + 4)^2 + (y + 2)^2 = k \]

4. **Expand the squares**:
Expanding the squares gives:
\[ (x^2 - 6x + 9) + (y^2 - 8y + 16) + (x^2 + 8x + 16) + (y^2 + 4y + 4) = k \]

5. **Combine like terms**:
Combine all the terms:
\[ 2x^2 + 2y^2 - 6x + 8x - 8y + 4y + 9 + 16 + 16 + 4 = k \]
Simplifying this:
\[ 2x^2 + 2y^2 + 2x - 4y + 45 = k \]

6. **Divide the equation by 2**:
To simplify further, divide the entire equation by 2:
\[ x^2 + y^2 + x - 2y + \frac{45}{2} = \frac{k}{2} \]

7. **Complete the square**:
We will complete the square for \( x \) and \( y \):
\[ x^2 + x + \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right)^2 + y^2 - 2y + 1 - 1 + \frac{45}{2} = \frac{k}{2} \]
This gives:
\[ \left(x + \frac{1}{2}\right)^2 + (y - 1)^2 + \frac{45}{2} - \frac{1}{4} - 1 = \frac{k}{2} \]

8. **Simplify the constant terms**:
Combine the constants:
\[ \frac{45}{2} - \frac{1}{4} - 1 = \frac{90}{4} - \frac{1}{4} - \frac{4}{4} = \frac{85}{4} \]
Hence, we have:
\[ \left(x + \frac{1}{2}\right)^2 + (y - 1)^2 = \frac{k}{2} - \frac{85}{4} \]

9. **Condition for a circle**:
For the equation to represent a circle, the right-hand side must be positive:
\[ \frac{k}{2} - \frac{85}{4} > 0 \implies k > \frac{85}{2} = 42.5 \]

10. **Determine the value of \( k \)**:
The options given are 30, 40, 55, and 35. The only value that satisfies \( k > 42.5 \) is:
\[ k = 55 \]

### Final Answer:
Thus, \( k \) is equal to \( 55 \).