Complex numbers and reflection

[erisedk]

Homework Statement

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

Homework Equations

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.

 

[haruspex]

Homework Statement

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

Homework Equations

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.

 

[haruspex]

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.

 

[ehild]

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?

 

[haruspex]

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.

If you have an equation for z that specifies a point, z=a say, then how do you write the equation for the reflection of that point? You would write $\bar z=a$ or $z=\bar a$, not $\bar z=\bar a$.

 

[erisedk]

Ok, got it thank you!