Question

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

• A. $x-iy=\frac{1-i}{1+2i}$
• B. $x+iy=\frac{1-i}{1-2i}$
• C. $x=\frac{3}{5}$
• D. $y=\frac{1}{5}$

Answer 1

Answer: (C), (D)

$y=\frac{1}{5}$

Explanation for the correct option.

Step 1: Write the conjugate of $(x+iy)(1-2i)$

Now conjugate is $\mathbf{(}\mathit{x}\mathbf{-}\mathit{i}\mathit{y}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{2}\mathit{i}\mathbf{)}$

According to the given information
$$\begin{gathered} \Rightarrow (\mathrm{x}-\mathrm{iy})(1+2\mathrm{i})=1+\mathrm{i} \\ \Rightarrow \mathrm{x}-\mathrm{iy}=\frac{1+\mathrm{i}}{1+2\mathrm{i}} \\ \Rightarrow \mathrm{x}-\mathrm{iy}=\frac{(1+\mathrm{i})(1-2\mathrm{i})}{5} \\ \Rightarrow \mathrm{x}-\mathrm{iy}=\frac{3-\mathrm{i}}{5} \end{gathered}$$
Step 2: Comparing real and imaginary parts on both sides

$\Rightarrow \mathrm{x}-\mathrm{iy}=\frac{3}{5}-\frac{i}{5}$
$\Rightarrow x=\frac{3}{5},y=\frac{1}{5}$

Hence, options C & D are correct.