When the problem says that the complex number $z$ moves on the straight line $y=2x$,what "clue" do I get from that? And generally when it says that a complex number belongs/moves to a conic section which thing should I considerate? My full problem is the following:
We have : $w=z+1/z$
$z$ moves on : $y=2x$
Find the locus (I am not sure if this is the correct word-I am foreigner) of $w$ /where $w$ moves.
Thanks in advance.
That a complex number $z$ moves along a line means that
$$z=z_1 t + z_2$$
where $t$ is real, $z_2$ is on the line, and $z_1$ (or rather the line from the origin to $z_1$) is parallel to the line.
In your case lets pick a point on the line $y=2x$. It could be $z_2=(0,0)$. Since the line passes through the origin, we can take $z_1$ to be any other point on the line, sat $z_1=(1,2)$.
We can compute that
$$z=t(1,2)+(0,0)=(t,2t).$$
Let us put this in the equation for $w$.
We compute
$$\begin{align}w &=z+1/z\\&=(t,2t)+1/(t,2t)\\&=(t,2t)+\left(\frac{t}{t^2+4t^2},\frac{-2t}{t^2+4t^2}\right)\\&=(t,2t)+\left(\frac{1}{5t},-\frac{2}{5t}\right)\\&=\left(t+\frac{1}{5t},2t-\frac{2}{5t}\right)\end{align}$$
What happens now, if we square the real and imaginary parts of this number and we (conveniently) subtract?
Let $R:=t+\frac{1}{5t}$ and $I:=2\left(t-\frac{1}{5t}\right)$, be the real and imaginary parts of $w$. Let us compute $4R^2-I^2$.
We get
$$4R^2-I^2=16/5, \ \ \ \ \ \ \text{ I think}.$$
Do you recognize this equation?
$$4x^2-y^2=16/5$$