Let the complex numbers `Z_(1), Z_(2) and Z_(3)` are the vertices A, B and C respectively of an isosceles right - angled triangle ABC with right angle at C, then the value of `((Z_(1)-Z_(2))^(2))/((Z_(1)-Z_(3))(Z_(3)-Z_(2)))` is equal to

Options

A. 2
B. 1
C. `-2`
D. `-1`

Answer

To solve the problem, we need to find the value of the expression:

\[ \frac{(Z_1 - Z_2)^2}{(Z_1 - Z_3)(Z_3 - Z_2)} \]

where \(Z_1\), \(Z_2\), and \(Z_3\) are the complex numbers representing the vertices of an isosceles right-angled triangle \(ABC\) with the right angle at \(C\).

### Step-by-Step Solution:

1. **Understanding the Triangle Configuration**:
- Since triangle \(ABC\) is isosceles and right-angled at \(C\), we can denote the lengths of the equal sides as \(a\). Thus, we have:
- \(AC = BC = a\)
- \(AB = a\sqrt{2}\) (the hypotenuse).

2. **Assigning Complex Numbers**:
- Let:
- \(Z_3 = Z_C = 0\) (the right angle vertex at the origin),
- \(Z_1 = Z_A = a\) (along the real axis),
- \(Z_2 = Z_B = ai\) (along the imaginary axis).

3. **Calculating the Differences**:
- Now, we calculate the differences:
- \(Z_1 - Z_2 = a - ai = a(1 - i)\)
- \(Z_1 - Z_3 = a - 0 = a\)
- \(Z_3 - Z_2 = 0 - ai = -ai\)

4. **Substituting into the Expression**:
- Substitute these values into the expression:
\[ \frac{(Z_1 - Z_2)^2}{(Z_1 - Z_3)(Z_3 - Z_2)} = \frac{(a(1 - i))^2}{(a)(-ai)} \]

5. **Calculating the Numerator**:
- The numerator becomes:
\[ (a(1 - i))^2 = a^2(1 - 2i + i^2) = a^2(1 - 2i - 1) = -2a^2i \]

6. **Calculating the Denominator**:
- The denominator becomes:
\[ (a)(-ai) = -a^2i \]

7. **Final Calculation**:
- Now substituting back into the expression:
\[ \frac{-2a^2i}{-a^2i} = 2 \]

Thus, the value of the expression is:

\[ \boxed{2} \]