Let $f(z)=az+b\overline{z}$, where $|a|>|b|$. Proof that $f(S^1)$ (the unit circunference) is an ellipse cantered at the origin, with the following characteristics:
i) The measure of the major semi-axis is $|a|+|b|$ and the minor's is $|a|-|b|$.
ii) If $b\neq 0$, $a=|a|(\cos \alpha+i\sin\alpha)$ and $b=|b|(\cos \beta+i\sin\beta)$m then the angle betwenn the semi-major axis and the real axis is$\frac{\alpha+\beta}{2}$.
I am really struglling to solve this question. I tried to write $z=x+iy$ and apply the definition of $f$, but I did a bunch of calculations and got nowhere. I don't see where I could apply the fact that $x^2+y^2=1$, because the expression of $f(x+iy)$ has only first degree terms.
Any help would be welcome.
first, notice that in point ii $a$ and $b$ are complex numbers too, so you'll need to remember that later. the way you apply $x^2+y^2=1$ is by finding the inverse function, then you apply the equation to that. You should get
$\left(\frac{x_1}{a+b}\right)^2+\left(\frac{y_1}{a-b}\right)^2=1$
(where $f(z)=x_1+iy_1$), which would be the equation of an ellipse with its major and minor axes aligned with the $x$ and $y$ axes, if $a$ and $b$ were real. Try to work from there, I hope this was helpful!