If the conjugate of `(x+iy)(1-2i)` be `1+i`, then

Options

A. `x=(1)/(5)`
B. `x+iy=(1)/(5)(3+i)`
C. `x-iy=(1)/(5)(3+i)`
D. `x+iy=(1-i)/(1+2i)`

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) such that the conjugate of \( (x + iy)(1 - 2i) \) equals \( 1 + i \).

### Step-by-Step Solution:

1. **Write the expression and its conjugate**:
We start with the expression:
\[ (x + iy)(1 - 2i) \]
The conjugate of this expression is given to be:
\[ \overline{(x + iy)(1 - 2i)} = 1 + i \]

2. **Multiply the expression**:
First, we multiply \( (x + iy) \) and \( (1 - 2i) \):
\[ (x + iy)(1 - 2i) = x(1 - 2i) + iy(1 - 2i) = x - 2xi + iy - 2y(-1) \]
This simplifies to:
\[ x + 2y + (y - 2x)i \]

3. **Write the conjugate**:
The conjugate of \( x + 2y + (y - 2x)i \) is:
\[ x + 2y - (y - 2x)i = x + 2y - yi + 2xi \]

4. **Set the conjugate equal to \( 1 + i \)**:
Now we equate this to \( 1 + i \):
\[ x + 2y = 1 \quad \text{(real part)} \]
\[ 2x - y = 1 \quad \text{(imaginary part)} \]

5. **Solve the system of equations**:
We have the following system of equations:
\[ x + 2y = 1 \quad \text{(1)} \]
\[ 2x - y = 1 \quad \text{(2)} \]

From equation (1), we can express \( x \) in terms of \( y \):
\[ x = 1 - 2y \]

Substitute \( x \) in equation (2):
\[ 2(1 - 2y) - y = 1 \]
Simplifying this gives:
\[ 2 - 4y - y = 1 \]
\[ 2 - 5y = 1 \]
\[ 5y = 1 \implies y = \frac{1}{5} \]

6. **Substitute \( y \) back to find \( x \)**:
Now substitute \( y \) back into the equation for \( x \):
\[ x = 1 - 2\left(\frac{1}{5}\right) = 1 - \frac{2}{5} = \frac{3}{5} \]

7. **Final result**:
Thus, we have:
\[ x = \frac{3}{5}, \quad y = \frac{1}{5} \]
Therefore, the expression \( x + iy \) can be written as:
\[ x + iy = \frac{3}{5} + i\frac{1}{5} \]