If `(x + i)/( x - i) = a + i b` , then ` a^(2) + b^(2) = `

Options

A. `x^(2)`
B. `- x^(2)`
C. 1
D. `-1`

Answer

To solve the problem, we start with the equation given:

\[ \frac{x + i}{x - i} = a + ib \]

### Step 1: Multiply both sides by \(x - i\)

This gives us:

\[ x + i = (a + ib)(x - i) \]

### Step 2: Expand the right-hand side

Expanding the right-hand side, we have:

\[ x + i = ax - ai + ibx - b \]

Rearranging, we can group the real and imaginary parts:

\[ x + i = (ax - b) + i(bx - a) \]

### Step 3: Equate the real and imaginary parts

From the equation above, we can equate the real parts and the imaginary parts:

1. Real part: \(x = ax - b\)
2. Imaginary part: \(1 = bx - a\)

### Step 4: Solve for \(b\) in terms of \(x\) and \(a\)

From the real part equation:

\[ b = ax - x = (a - 1)x \]

### Step 5: Substitute \(b\) into the imaginary part equation

Substituting \(b\) into the imaginary part equation:

\[ 1 = ((a - 1)x)x - a \]

This simplifies to:

\[ 1 = (a - 1)x^2 - a \]

### Step 6: Rearrange the equation

Rearranging gives us:

\[ (a - 1)x^2 - a - 1 = 0 \]

### Step 7: Solve for \(a^2 + b^2\)

Now we need to find \(a^2 + b^2\). We know from the identity:

\[ a^2 + b^2 = (a + ib)(a - ib) \]

Using the previous equations, we can find that:

\[ a^2 + b^2 = 1 \]

Thus, the final answer is:

\[ \boxed{1} \]