Question

Complex number #1

Find the set of complex number such that |\(\frac{z-3}{z+3}\)|=2
How to start?

Answer 1

If I remember correctly then:
\[\Large z\cdot z^*=|z| \]
Might want to complex conjugate anyway, for the denominator's sake.

Answer 2

\[z^*\]is the conjugate?

Answer 3

If so, then it should be \[z\cdot z^* = |z|^2\]

Answer 4

Yes you're right @Callisto, I forgot the square.

Answer 5

z = x+yi
\[|z -3| = r_1=\sqrt{(x-3)^2+y^2}\]\[|z +3| = r_2=\sqrt{(x+3)^2+y^2}\]
\[|z-3| = |r_1e^{i\theta_1}|\]\[|z+3| = |r_2e^{i\theta_2}|\]
\[|\frac{z-3}{z+3}|=\frac{r_1}{r_2}\]
Then, some substitution work, we get
(x+5)^2 + y^2 = 16
So, the set is {z : z=x+yi, (x+5)^2 + y^2 = 16}

Right? Wrong? Any other methods?

Answer 6

http://www.wolframalpha.com/input/?i= |%28z-3%29%2F%28z%2B3%29|%3D2

looks like it's right