Q120 / 180 · COMEDK 2015 · Mathematics

If the conjugate of $(x+i y)(1-2 i)$ is $1+i$, then

  • A
    $x-i y=\frac{1-i}{1-2 i}$
  • Correct answer
    B
    $x+i y=\frac{1-i}{1-2 i}$
  • C
    $x=\frac{1}{5}$
  • D
    $x=-\frac{1}{5}$

Solution

We have, $\overline{(x+i y)(1-2 i)}=1+i$ $\Rightarrow \quad \overline{(x+i y)(1-2 i)}=1+i$ $\Rightarrow \quad(x-i y)(1+2 i)=1+i$ $\Rightarrow \quad x-i y=\frac{1+i}{1+2 i}$ $\Rightarrow \quad \overline{x-i y}=\overline{\left(\frac{1+i}{1+2 i}\right)}=\frac{1-i}{1-2 i}$ $\Rightarrow \quad x+i y=\frac{1-i}{1-2 i}$

Hence, the correct answer is option: 'B'.

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