Complex Numbers

[Pranav-Arora]

1. The problem statement, all variables and given/known data
If \frac{z_1-2z_2}{2-z_1\overline{z}_2} is unimodulus and z2 is not unimodulus, then find |z1|.


2. Relevant equations



3. 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.
|\frac{z_1-2z_2}{2-z_1\overline{z}_2}|=1
and
|z_2|≠1

[vela]

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

EDIT: Also |w/z| = 1 means |w|=|z|.

[Pranav-Arora]

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

Thanks for the reply vela! :smile:

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

But now i am stuck here. :(

[Simon Bridge]

Thanks for the reply vela! :smile:

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

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?

[Pranav-Arora]

:) 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? :)

[Pranav-Arora]

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. :)

[Simon Bridge]

Well done!

For dessert:

z = \left ( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i \right )^{12}
...rewrite z in it's simplest form (it will be exact).

[Pranav-Arora]

Well done!

For dessert:

z = \left ( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i \right )^{12}
...rewrite z in it's simplest form (it will be exact).

z=-1. :)

[Simon Bridge]

Sweet dessert: That one is usually nasty because everyone tries to brute-force it. :)