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 will find the coordinates of points C and D in the complex plane, given points A and B, and then calculate the required expression.

### Step 1: Identify the coordinates of points A and B
We have:
- \( A = 2 + 3i \)
- \( B = 3 + 4i \)

### Step 2: Calculate the vector AB
The vector \( \overrightarrow{AB} \) can be calculated as:
\[ \overrightarrow{AB} = B - A = (3 + 4i) - (2 + 3i) = 1 + i \]

### Step 3: Find the length of AB
The length of \( AB \) is given by:
\[ |AB| = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2} \]

### Step 4: Determine the coordinates of point D
Since ABCD is a square, the angle between the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AD} \) is \( 90^\circ \). To find point D, we can rotate the vector \( \overrightarrow{AB} \) by \( 90^\circ \) in the anticlockwise direction. This can be done by multiplying \( \overrightarrow{AB} \) by \( i \):
\[ \overrightarrow{AD} = i \cdot \overrightarrow{AB} = i(1 + i) = i + i^2 = i - 1 = -1 + i \]
Thus, point D can be found as:
\[ D = A + \overrightarrow{AD} = (2 + 3i) + (-1 + i) = (2 - 1) + (3 + 1)i = 1 + 4i \]

### Step 5: Determine the coordinates of point C
Similarly, we can find point C by rotating \( \overrightarrow{AB} \) by \( 90^\circ \) in the clockwise direction:
\[ \overrightarrow{AC} = -i \cdot \overrightarrow{AB} = -i(1 + i) = -i - i^2 = -i + 1 = 1 - i \]
Thus, point C can be found as:
\[ C = A + \overrightarrow{AC} = (2 + 3i) + (1 - i) = (2 + 1) + (3 - 1)i = 3 + 2i \]

### Step 6: Calculate the moduli of Z3 and Z4
Now we can find the moduli of points C and D:
- For \( Z_3 = C = 3 + 2i \):
\[ |Z_3| = |3 + 2i| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \]
- For \( Z_4 = D = 1 + 4i \):
\[ |Z_4| = |1 + 4i| = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \]

### Step 7: Calculate \( |Z_3|^2 - |Z_4|^2 \)
Now we compute:
\[ |Z_3|^2 - |Z_4|^2 = (\sqrt{13})^2 - (\sqrt{17})^2 = 13 - 17 = -4 \]

### Final Answer
Thus, the value of \( |Z_3|^2 - |Z_4|^2 \) is:
\[ \boxed{-4} \]