If `(x-yi)+(a+bi)=2x and i=sqrt(-1)`, then `(x+yi)(a+bi)=`

Options

A. `x^(2)+y^(2)`
B. `x^(2)-y^(2)`
C. `4x^(2)+y^(2)`
D. `5x^(2)`

Answer

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

\[ (x - yi) + (a + bi) = 2x \]

### Step 1: Separate Real and Imaginary Parts
We can rewrite the equation by combining the real and imaginary parts:

\[ (x + a) + (y + b)i = 2x \]

### Step 2: Compare Real and Imaginary Parts
From the equation above, we can compare the real and imaginary parts:

1. Real part: \( x + a = 2x \)
2. Imaginary part: \( y + b = 0 \)

### Step 3: Solve for \(a\) and \(b\)
From the real part equation:

\[ a = 2x - x = x \]

From the imaginary part equation:

\[ b = -y \]

### Step 4: Substitute \(a\) and \(b\) into the Expression
Now we need to find the value of \((x + yi)(a + bi)\):

\[ (x + yi)(a + bi) = (x + yi)(x - yi) \]

### Step 5: Expand the Expression
Using the distributive property (FOIL method):

\[ = x^2 + x(-yi) + yi(x) + yi(-yi) \]

This simplifies to:

\[ = x^2 - xy i + xy i - y^2 i^2 \]

### Step 6: Simplify Using \(i^2 = -1\)
Since \(i^2 = -1\), we can replace \( -y^2 i^2\) with \(y^2\):

\[ = x^2 + y^2 \]

### Final Result
Thus, we find that:

\[ (x + yi)(a + bi) = x^2 + y^2 \]

### Conclusion
The final answer is:

\[ \boxed{x^2 + y^2} \]