If `x+ iy=(1+ 4i)(1+5i)`, then `(x^(2)+y^(2))` is equal to :

Options

A. 17
B. 26
C. 442
D. None of these

Answer

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

\[ x + iy = (1 + 4i)(1 + 5i) \]

### Step 1: Multiply the complex numbers
We need to multiply the two complex numbers on the right side:

\[ (1 + 4i)(1 + 5i) = 1 \cdot 1 + 1 \cdot 5i + 4i \cdot 1 + 4i \cdot 5i \]

### Step 2: Simplify the multiplication
Calculating each term:

- \( 1 \cdot 1 = 1 \)
- \( 1 \cdot 5i = 5i \)
- \( 4i \cdot 1 = 4i \)
- \( 4i \cdot 5i = 20i^2 = 20(-1) = -20 \)

Now, combine these results:

\[ 1 + 5i + 4i - 20 = 1 - 20 + (5i + 4i) = -19 + 9i \]

### Step 3: Identify the real and imaginary parts
From the expression \( x + iy = -19 + 9i \), we can identify:

- Real part \( x = -19 \)
- Imaginary part \( y = 9 \)

### Step 4: Calculate \( x^2 + y^2 \)
Now we need to find \( x^2 + y^2 \):

\[ x^2 + y^2 = (-19)^2 + 9^2 \]

Calculating each square:

- \( (-19)^2 = 361 \)
- \( 9^2 = 81 \)

Adding these together:

\[ x^2 + y^2 = 361 + 81 = 442 \]

### Final Answer
Thus, the value of \( x^2 + y^2 \) is:

\[ \boxed{442} \]