If the point $z_1=1+i$ where $i=\sqrt{-1}$ is the reflection of $a$ point $z_2=x+i y$ in the line $i \bar{z}-i z=5$, then the point $z_2$ is

• Correct answer
  A

  $1+4 \mathrm{i}$
• B

  $4+ \mathrm{i}$
• C

  $1- \mathrm{i}$
• D

  $-1- \mathrm{i}$

Solution

Let $\mathrm{z}=\mathrm{a}+\mathrm{bi}$
$\begin{aligned} & \Rightarrow \mathrm{z}=\mathrm{a}-\mathrm{bi} \\ & \therefore \mathrm{i} \overline{\mathrm{z}}-\mathrm{iz}=\mathrm{i}[(\mathrm{a}-\mathrm{bi})-(\mathrm{a}+\mathrm{bi})]=5 \\ & \Rightarrow \mathrm{i}[-2 \mathrm{bi}]=5 \\ & \Rightarrow \mathrm{~b}=\frac{5}{2} \end{aligned}$ So from figure it is clear that $\begin{aligned} & \mathrm{x}=1, \mathrm{y}=\frac{5}{2}+\frac{3}{2}=4 \\ & \mathrm{z}_2=1+4 \mathrm{i} \end{aligned}$

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