Two different demonstrations on the complex plane

[Telemachus]

Hi there, I have to prove this two sentences . I think I've solved the first, but I'm quiet stuck with the second.

The first says:

1) Demonstrate that the equation of a line or a circumference in the complex plane can be written this way: $$\alpha z . \bar{z}+\beta z+\bar{\beta} \bar{z}+\gamma=0$$, with $$\alpha,\gamma\in{R},\beta\in{C}$$

So I called z and beta:

$$z=x+iy,\beta=u+iv$$

Then developing the products I get:
$$\alpha(x^2+y^2)+2(ux-vy)+\gamma=0$$

And making alpha equal zero I get the equation for a line, right? (for u and v fixed).

Then completing the square and reordering:

$$\left(x+\displaystyle\frac{u}{\alpha}\right )^2+\left(y-\displaystyle\frac{v}{\alpha}\right )^2=\displaystyle\frac{u^2}{\alpha^2}+\frac{v^2}{\alpha^2}-\frac{\gamma}{\alpha}$$

This is the equation for the circle, is this right?

In the other hand I got:

2) Prove that the geometrical place for the points $$z\in{C}$$ that verifies $$\left\|{\displaystyle\frac{z-1}{z+1}}\right\|=k$$ is a circumference ($$k\in{R},1\neq{k}>0$$).

I couldn't make much for this. I called z=x+iy again:

$$\left\|{\frac{z-1}{z+1}}\right\|=k\Rightarrow{\left\|{\frac{(x+iy-1)(x-iy+1)}{(x+iy+1)(x-iy+1)}}\right\|=k}\Rightarrow{\left\|{\frac{x^2+y^2+2iy-1}{(x+1)^2+y^2}}\right\|=k}$$

I don't know what to do from there.

Bye.

PS: sorry if it bothers you that I made two questions in the same topic, but I thought it was quiet trivial and it didn't deserved two different topics, but I can split it if necessary.

 

[Dickfore]

The thing in the denominator is real by construction (You multiplied it by the complex conjugate). What are the real and imaginary parts of the numerator?

 

[Telemachus]

$$Re(w)=\frac{x^2+y^2-1}{(x+1)^2+y^2}$$
$$Im(w)=\frac{2y}{(x+1)^2+y^2}$$

BTW, is the first exercise right?

 

[Dickfore]

The first exercise is correct.

For the second one, a modulus of a complex number:
$$|w| = \sqrt{ \left( \mathrm{Re} w \right)^{2} + \left( \mathrm{Im} w \right)^{2}} = k$$
Square the equation, get rid off the denominators and simplify the expression. Write what you get.

 

[Telemachus]

Thanks :)

 

[Dickfore]

Actually, it does not seem to correspond to a circumference. The condition

$$\left|\frac{z - 1}{z + 1}\right| = k \Leftrightarrow |z - 1| = k |z + 1|$$

means that the distance from the point $(1, 0)$ is $k$ times bigger than the distance from the point $(-1, 0)$. I am not sure if this is a circle in general.

 

[Dickfore]

Actually, scratch what I was saying. Square the above:
$$|z - 1|^{2} = k^{2} |z + 1|^{2}$$

But, $| w |^{2} = \bar{w} w$, so:
$$(z - 1)(\bar{z} - 1) = k^{2} (z + 1) (\bar{z} + 1)$$

Simplify this and use part 1 of the problem.

 

[Telemachus]

Thank you very much :)