If ` a^(2)+ b^(2) = 2 and c^(2) + d^(2) = 1 ` then the value of `(ad - bc)^(2) + (ac + bd)^(2)` is

Options

A. `(4)/(9)`
B. `(1)/(2)`
C. 1
D. 2

Answer

To solve the problem, we need to evaluate the expression \((ad - bc)^{2} + (ac + bd)^{2}\) given the equations \(a^{2} + b^{2} = 2\) and \(c^{2} + d^{2} = 1\).

### Step-by-Step Solution:

1. **Understanding the Given Equations:**
- From the equation \(a^{2} + b^{2} = 2\), we can deduce that the maximum values for \(a\) and \(b\) can be \(1\) each, since \(1^{2} + 1^{2} = 2\).
- Therefore, we can set \(a = 1\) and \(b = 1\).

2. **Analyzing the Second Equation:**
- The equation \(c^{2} + d^{2} = 1\) suggests that \(c\) and \(d\) must be such that their squares add up to \(1\).
- A simple choice is \(c = 0\) and \(d = 1\) (since \(0^{2} + 1^{2} = 1\)).

3. **Substituting Values into the Expression:**
- Now we substitute \(a = 1\), \(b = 1\), \(c = 0\), and \(d = 1\) into the expression \((ad - bc)^{2} + (ac + bd)^{2}\).
- First, calculate \(ad - bc\):
\[ ad - bc = (1)(1) - (1)(0) = 1 - 0 = 1 \]
- Now, calculate \((ad - bc)^{2}\):
\[ (ad - bc)^{2} = (1)^{2} = 1 \]

4. **Calculating the Second Part of the Expression:**
- Next, calculate \(ac + bd\):
\[ ac + bd = (1)(0) + (1)(1) = 0 + 1 = 1 \]
- Now, calculate \((ac + bd)^{2}\):
\[ (ac + bd)^{2} = (1)^{2} = 1 \]

5. **Final Calculation:**
- Combine the results from both parts:
\[ (ad - bc)^{2} + (ac + bd)^{2} = 1 + 1 = 2 \]

### Conclusion:
The value of \((ad - bc)^{2} + (ac + bd)^{2}\) is \(2\).