Let `z_1=2+3i` and `z_2=3+4i`. The set `S={z in C:|z-z_1|^(2)-|z-z_2|^2=|z_1-z_2|^2}` represents a

Options

A. straight line with the sum of its intercepts on the coordinate axes equals –18
B. hyperbola with eccentricity 2
C. hyperbola with the length of the transverse axis 7
D. straight line with the sum of its intercepts on the coordinate axes equal to 14

Answer

To solve the problem, we need to analyze the expression given in the set \( S = \{ z \in \mathbb{C} : |z - z_1|^2 - |z - z_2|^2 = |z_1 - z_2|^2 \} \).

### Step-by-Step Solution:

1. **Identify the Complex Numbers**:
We have \( z_1 = 2 + 3i \) and \( z_2 = 3 + 4i \).

2. **Calculate \( |z_1 - z_2|^2 \)**:
\[ z_1 - z_2 = (2 + 3i) - (3 + 4i) = -1 - i \]
\[ |z_1 - z_2|^2 = |-1 - i|^2 = (-1)^2 + (-1)^2 = 1 + 1 = 2 \]

3. **Rewrite the Set Condition**:
The condition becomes:
\[ |z - z_1|^2 - |z - z_2|^2 = 2 \]

4. **Express \( |z - z_1|^2 \) and \( |z - z_2|^2 \)**:
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers.
\[ |z - z_1|^2 = |(x - 2) + (y - 3)i|^2 = (x - 2)^2 + (y - 3)^2 \]
\[ |z - z_2|^2 = |(x - 3) + (y - 4)i|^2 = (x - 3)^2 + (y - 4)^2 \]

5. **Substitute into the Condition**:
\[ (x - 2)^2 + (y - 3)^2 - \left[(x - 3)^2 + (y - 4)^2\right] = 2 \]

6. **Expand Both Sides**:
Expanding the left side:
\[ (x^2 - 4x + 4 + y^2 - 6y + 9) - (x^2 - 6x + 9 + y^2 - 8y + 16) = 2 \]
Simplifying:
\[ -4x + 4 + 6x - 9 + 8y - 16 = 2 \]
\[ 2x + 2y - 21 = 2 \]

7. **Rearranging the Equation**:
\[ 2x + 2y = 23 \implies x + y = \frac{23}{2} \]

8. **Final Form of the Equation**:
The equation \( x + y = \frac{23}{2} \) represents a straight line in the complex plane.

### Conclusion:
The set \( S \) represents a straight line in the complex plane.