If `A(2+3i) and B(3+4i)` are two vertices of a square ABCD (taken in anticlockwise order)in a complex plane, then the value of `|Z_(3)|^(2)-|Z_(4)|^(2)` (Where C is `Z_(3)` and D is `Z_(4)`) is equal to

Options

A. 0
B. 6
C. 8
D. 12

Answer

To solve the problem, we need to find the values of the vertices C and D of the square ABCD given the vertices A and B in the complex plane. We will then compute the expression \( |Z_3|^2 - |Z_4|^2 \), where \( Z_3 \) corresponds to vertex C and \( Z_4 \) corresponds to vertex D.

### Step-by-Step Solution:

1. **Identify the given points**:
- Let \( A = 2 + 3i \)
- Let \( B = 3 + 4i \)

2. **Calculate the vector AB**:
\[ AB = B - A = (3 + 4i) - (2 + 3i) = 1 + i \]

3. **Find the length of AB**:
\[ |AB| = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2} \]

4. **Determine the direction of the perpendicular vector**:
Since the angle between sides of a square is \( 90^\circ \), we can rotate the vector \( AB \) by \( 90^\circ \) to find the vectors \( AC \) and \( AD \). The rotation can be achieved by multiplying by \( i \):
\[ AC = i \cdot AB = i(1 + i) = i + i^2 = i - 1 = -1 + i \]

5. **Calculate the coordinates of point C**:
\[ C = A + AC = (2 + 3i) + (-1 + i) = (2 - 1) + (3 + 1)i = 1 + 4i \]
Thus, \( Z_3 = 1 + 4i \).

6. **Find the other perpendicular vector**:
The other perpendicular vector can be obtained by rotating \( AB \) in the opposite direction:
\[ AD = -i \cdot AB = -i(1 + i) = -i - i^2 = -i + 1 = 1 - i \]

7. **Calculate the coordinates of point D**:
\[ D = A + AD = (2 + 3i) + (1 - i) = (2 + 1) + (3 - 1)i = 3 + 2i \]
Thus, \( Z_4 = 3 + 2i \).

8. **Calculate the magnitudes of Z3 and Z4**:
\[ |Z_3|^2 = |1 + 4i|^2 = 1^2 + 4^2 = 1 + 16 = 17 \]
\[ |Z_4|^2 = |3 + 2i|^2 = 3^2 + 2^2 = 9 + 4 = 13 \]

9. **Compute the final expression**:
\[ |Z_3|^2 - |Z_4|^2 = 17 - 13 = 4 \]

### Final Answer:
The value of \( |Z_3|^2 - |Z_4|^2 \) is \( 4 \).