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Complex numbers and reflection

  1. Apr 27, 2016 #1
    1. The problem statement, all variables and given/known data
    Reflection of the line ##\bar{a}z + a\bar{z} = 0## in the real axis is

    2. Relevant equations


    3. 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.
     
  2. jcsd
  3. Apr 27, 2016 #2

    haruspex

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    What is the reflection of z in the real axis?
     
  4. Apr 27, 2016 #3
    Its conjugate.
     
  5. Apr 27, 2016 #4

    haruspex

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    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?
     
  6. Apr 27, 2016 #5
    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.
     
  7. Apr 27, 2016 #6

    ehild

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    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?
     
  8. Apr 27, 2016 #7

    haruspex

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    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##.
     
  9. Apr 28, 2016 #8
    Ok, got it thank you!
     
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