# Complex numbers and reflection

erisedk

## Homework Statement

Reflection of the line ##\bar{a}z + a\bar{z} = 0## in the real axis is

## The Attempt at a Solution

I know that a line in the complex plane is represented as ##\bar{a}z + a\bar{z} + b= 0## and that its slope ##μ = \dfrac{-a}{\bar{a}}##. I'm not sure how to do this problem. I'm also not very good with complex geometry so please help.

Homework Helper
Gold Member

## Homework Statement

Reflection of the line ##\bar{a}z + a\bar{z} = 0## in the real axis is

## The Attempt at a Solution

I know that a line in the complex plane is represented as ##\bar{a}z + a\bar{z} + b= 0## and that its slope ##μ = \dfrac{-a}{\bar{a}}##. I'm not sure how to do this problem. I'm also not very good with complex geometry so please help.
What is the reflection of z in the real axis?

erisedk
Its conjugate.

Homework Helper
Gold Member
Its conjugate.
Right. So if you have two points ##z## and ##\bar w##, how would you write their reflections notationally? What is the general rule you see here?

erisedk
As ##\bar{z}## and ##w##? That I need to take the conjugate of the equation of the line? But that gives me back the original line. However, the funny thing is if I take the conjugate of only ##z##, I get the desired answer, i.e. ##\bar{a}\bar{z} + az = 0##. I can't really explain that though.

Homework Helper
However, the funny thing is if I take the conjugate of only ##z##, I get the desired answer, i.e. ##\bar{a}\bar{z} + az = 0##. I can't really explain that though.
You need to conjugate those complex numbers z which are on that line instead of conjugating the equation.
You have a line in the x,y plane. What is the equation of that line?
What line do you get when you reflect the original line on the x axis?
How can you write the complex numbers z1 with their real and imaginary parts which are on the original line ? What are those complex numbers z2 which are on the reflected line?