If `a ^(2) + b ^(2) + (1)/(a ^(2)) + (1)/(b ^(2)) =4,` then the value of `a ^(2) + b ^(2) ` will be

Options

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

Answer

To solve the equation \( a^2 + b^2 + \frac{1}{a^2} + \frac{1}{b^2} = 4 \) and find the value of \( a^2 + b^2 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ a^2 + b^2 + \frac{1}{a^2} + \frac{1}{b^2} = 4 \]

### Step 2: Introduce new variables
Let \( x = a^2 \) and \( y = b^2 \). The equation then becomes:
\[ x + y + \frac{1}{x} + \frac{1}{y} = 4 \]

### Step 3: Combine terms
We can rewrite the equation as:
\[ x + y + \frac{1}{x} + \frac{1}{y} = x + y + \frac{y + x}{xy} = 4 \]
This simplifies to:
\[ x + y + \frac{x + y}{xy} = 4 \]

### Step 4: Let \( s = x + y \) and \( p = xy \)
Now, substituting \( s \) and \( p \):
\[ s + \frac{s}{p} = 4 \]
Multiplying through by \( p \) gives:
\[ sp + s = 4p \]
Rearranging this, we get:
\[ sp - 4p + s = 0 \]

### Step 5: Factor the equation
Factoring out \( s \):
\[ s(p + 1) = 4p \]
Thus, we have:
\[ s = \frac{4p}{p + 1} \]

### Step 6: Find values for \( s \) and \( p \)
To find \( s \), we can test values for \( p \). If we try \( p = 1 \):
\[ s = \frac{4 \cdot 1}{1 + 1} = \frac{4}{2} = 2 \]

### Step 7: Verify the values
If \( s = 2 \) and \( p = 1 \), we can check if \( x \) and \( y \) can be \( 1 \):
- \( x + y = 2 \)
- \( xy = 1 \)

The numbers \( x \) and \( y \) that satisfy these conditions are \( 1 \) and \( 1 \):
\[ x = 1, \quad y = 1 \]

### Step 8: Conclusion
Thus, \( a^2 + b^2 = x + y = 1 + 1 = 2 \).

The final answer is:
\[ \boxed{2} \]