Suppose a and b are two different complete numbers such that `|a + sqrt(a^(2) - b^(2))| + | a - sqrt(a^(2) - b^(2))| = |a + b| + 4` then |a-b| is equal to _________

Answer

To solve the problem, we need to analyze the given equation involving the complex numbers \( a \) and \( b \):

\[ |a + \sqrt{a^2 - b^2}| + |a - \sqrt{a^2 - b^2}| = |a + b| + 4 \]

### Step 1: Simplify the expressions
Let \( x = \sqrt{a^2 - b^2} \). Then we can rewrite the left-hand side of the equation as:

\[ |a + x| + |a - x| \]

### Step 2: Use the property of absolute values
Using the property of absolute values, we know:

\[ |a + x| + |a - x| = 2 \max(|a|, |x|) \]

This is because the sum of the absolute values of two terms can be expressed in terms of the maximum of those terms.

### Step 3: Rewrite the equation
Now, we can rewrite our equation as:

\[ 2 \max(|a|, |x|) = |a + b| + 4 \]

### Step 4: Analyze the term \( x = \sqrt{a^2 - b^2} \)
Since \( x \) is defined as \( \sqrt{a^2 - b^2} \), we need to consider the conditions under which this expression is valid. For \( x \) to be real, we require \( a^2 \geq b^2 \).

### Step 5: Consider cases for \( |a| \) and \( |x| \)
1. **Case 1**: If \( |a| \geq |x| \), then:

\[ 2|a| = |a + b| + 4 \]

2. **Case 2**: If \( |x| > |a| \), then:

\[ 2|x| = |a + b| + 4 \]

### Step 6: Solve for \( |a - b| \)
We will focus on the first case for simplicity. Rearranging gives us:

\[ |a| = \frac{|a + b| + 4}{2} \]

Now, we can express \( |a - b| \) in terms of \( |a| \) and \( |b| \):

Using the triangle inequality:

\[ |a - b| \leq |a| + |b| \]

And the reverse triangle inequality:

\[ |a - b| \geq ||a| - |b|| \]

### Step 7: Substitute and simplify
From our earlier equation, we can express \( |b| \):

Let \( |b| = |a| - 4 \) (assuming \( |a| \geq |b| \)). Then:

\[ |a - b| = |a| + |b| = |a| + (|a| - 4) = 2|a| - 4 \]

### Step 8: Final calculation
Substituting \( |a| = \frac{|a + b| + 4}{2} \) into our expression for \( |a - b| \):

\[ |a - b| = 2\left(\frac{|a + b| + 4}{2}\right) - 4 = |a + b| + 4 - 4 = |a + b| \]

### Conclusion
Thus, we find that:

\[ |a - b| = 4 \]