# Complex Numbers

1. Oct 28, 2011

### Saitama

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
$$|z_2|≠1$$

Last edited: Oct 28, 2011
2. Oct 28, 2011

### vela

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

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

3. Oct 28, 2011

### Saitama

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. :(

4. Oct 29, 2011

### Simon Bridge

x + 4y = 4 + xy

and you had to find x, what would you do?

5. Oct 29, 2011

### Saitama

I still dont understand. :(

Can you give me one more hint? :)

6. Oct 29, 2011

### Saitama

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. Oct 29, 2011

### 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).

8. Oct 29, 2011

### Saitama

z=-1. :)

9. Oct 29, 2011

### Simon Bridge

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