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 need to find \( x^2 + y^2 \) given that \( x + iy = (1 + 4i)(1 + 5i) \).

### Step-by-Step Solution:

1. **Multiply the Complex Numbers**:
\[ (1 + 4i)(1 + 5i) \]
We will use the distributive property (FOIL method):
\[ = 1 \cdot 1 + 1 \cdot 5i + 4i \cdot 1 + 4i \cdot 5i \]
\[ = 1 + 5i + 4i + 20i^2 \]

2. **Simplify the Expression**:
Recall that \( i^2 = -1 \):
\[ 20i^2 = 20(-1) = -20 \]
Now substituting this back:
\[ = 1 + 5i + 4i - 20 \]
Combine like terms:
\[ = (1 - 20) + (5i + 4i) = -19 + 9i \]

3. **Identify \( x \) and \( y \)**:
From the equation \( x + iy = -19 + 9i \), we can see that:
\[ x = -19 \quad \text{and} \quad y = 9 \]

4. **Calculate \( x^2 + y^2 \)**:
Now we need to find \( x^2 + y^2 \):
\[ x^2 + y^2 = (-19)^2 + 9^2 \]
Calculate each square:
\[ = 361 + 81 \]
\[ = 442 \]

### Final Answer:
Thus, \( x^2 + y^2 = 442 \).