Complex Numbers

  • Thread starter Saitama
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  • #1
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Homework Statement


If [itex]\frac{z_1-2z_2}{2-z_1\overline{z}_2}[/itex] is unimodulus and z2 is not unimodulus, then find |z1|.


Homework Equations





The Attempt at a Solution


I am a complete dumb at Complex numbers, please someone guide me in the right direction.
In this question, what i understand is this, and nothing else.
[tex]|\frac{z_1-2z_2}{2-z_1\overline{z}_2}|=1[/tex]
and
[tex]|z_2|≠1[/tex]
 
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Answers and Replies

  • #2
vela
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It should be [itex]|z_2| \ne 1[/itex] and
[tex]\left\lvert \frac{z_1-2z_2}{2-z_1\overline{z}_2} \right\rvert = 1[/tex]Square both sides and use the fact that [itex]|z|^2 = z\bar z[/itex].

EDIT: Also |w/z| = 1 means |w|=|z|.
 
  • #3
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92
It should be [itex]|z_2| \ne 1[/itex] and
[tex]\left\lvert \frac{z_1-2z_2}{2-z_1\overline{z}_2} \right\rvert = 1[/tex]Square both sides and use the fact that [itex]|z|^2 = z\bar z[/itex].
Thanks for the reply vela! :smile:

I squared both the sides and using the fact [itex]|z|^2 = z\bar z[/itex], i get:-
[tex]|z_1|^2+4|z_2|^2=4+|z_1|^2|z_2|^2[/tex]

But now i am stuck here. :(
 
  • #4
Simon Bridge
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Thanks for the reply vela! :smile:

I squared both the sides and using the fact [itex]|z|^2 = z\bar z[/itex], i get:-
[tex]|z_1|^2+4|z_2|^2=4+|z_1|^2|z_2|^2[/tex]

But now i am stuck here. :(
:) Follow your nose: if it was

x + 4y = 4 + xy

and you had to find x, what would you do?
 
  • #5
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92
:) Follow your nose: if it was

x + 4y = 4 + xy

and you had to find x, what would you do?
I still dont understand. :(

Can you give me one more hint? :)
 
  • #6
3,812
92
Thank you both for the help. I have figured it out. :)

x+4y=4+xy
or x-xy=4-4y
or x(1-y)=4(1-y)
or x=4
or |z1|=2.

Thanks again. :)
 
  • #7
Simon Bridge
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Well done!

For dessert:

[tex]z = \left ( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i \right )^{12}[/tex]
...rewrite z in it's simplest form (it will be exact).
 
  • #8
3,812
92
Well done!

For dessert:

[tex]z = \left ( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i \right )^{12}[/tex]
...rewrite z in it's simplest form (it will be exact).
z=-1. :)
 
  • #9
Simon Bridge
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Sweet dessert: That one is usually nasty because everyone tries to brute-force it. :)
 

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