Finding the locus of $z=at+\frac bt$

Question

I have to find the locus of$$ z=at+\frac{b}{t}, $$
where $a, b \in \mathbb{C}$ are constants. I took $a=a_1+ia_2$ and $b=b_1+ib_2$, but could not get the solution.

Answer 1

In a real setting, the curve given by parametric equations

$$\tag{1}x=t, \ \ \ y=\dfrac{1}{t},$$

i.e., with cartesian equation $y=\dfrac{1}{x}$, is a (equilateral) hyperbola.

Of course, (1) can be written

$$\tag{2}x+iy=t\color{red}{1}+\dfrac{1}{t}\color{red}{i}$$

Thus $z=t\color{red}{a}+\dfrac{1}{t}\color{red}{b}$ (see the analogy with (2)) is a hyperbola with respect to oblique axes defined by $\vec{OA}$ and $\vec{OB}$ ($A,B$ are points associated with complex numbers $a,b$). Thus it is also a hyperbola with respect to standard axes.