Question

Reflection of the complex number $\frac{2-i}{3+i}$ in the straight line $z(1+i)$ = $\overline{z}(i-1)$ is

• A. $\frac{-1+i}{2}$
• B. $\frac{i-1}{2}$
• C. $-i+1$
• D. $\frac{-i+1}{2}$

Answer 1

The correct option is A

$\frac{-1+i}{2}$

Let $z_1$ = $\frac{2-i}{3+i}$ = $\frac{(2-i)(3-i)}{(3+i)(3-i)}$
= $6-3i-2i+i^2$
= $\frac{5-5i}{10}$
= $\frac{1}{2}(1-i)$
The equation of the straight line is
Z(1 + i) = $\overline{z}(i-1)$
(z + $\overline{z}$) + i(z - $\overline{z}$) = 0
$2x+2iyi$ = 0
$\Rightarrow 2x-2y=0$
$\Rightarrow x-y$ = 0
Let the reflection of $z_1$ be $z_2$ ($z_1$ + i$y_1$)
To find the reflection about y = x, interchange the x and y co-ordinates
$\Rightarrow$The reflection of $\frac{1}{2}$ + $\frac{-1}{2}$i is
$\frac{-1}{2}$ + $\frac{1}{2}$i = $\frac{-1+i}{2}$