we know：ellipse is the locus of dynamic point whose distances from two fixed points’ sum is fixed value.
so I write: $z$ is complex number, and $a,b,c,t$ are reals.
$$\begin{align*}\left| z-a\right| +\left| z-b\right| =c\tag{1}\end{align*}$$
Is there any guidelines/rules for choosing the $t$ and $c$? or $2t$? $2$ maybe related with focal length sometimes.
$$\begin{align*}\left| z-a\right| +\left| z-b\right| =t\tag{2}\end{align*}$$
$$\begin{align*}\left| z-a\right| +\left| z-b\right| =2t\tag{3}\end{align*}$$
The choice between using a variable (such as $t$) or that variable multiplied by a constant (such as $2t$) usually depends on the expressions that occur later on in the computation.
For example, if you started with $t$, and $t/2$ occurred many times later on, you might prefer to start with $2t$ so that $t$ would occur many times.
Another example is Fourier integrals, where you can take the integral from $-\pi$ to $\pi$ and the $\sin$ and $\cos$ of $x$ or take the integral from $-1$ to $1$ and the $\sin$ and $\cos$ of $\pi x$.
Another example: when working out a $\delta-\epsilon$ proof, many books starta proof something like this: "Let $\epsilon = \delta/4$." After some computations, the result comes out "$|...| < \epsilon$." They seem unwilling to start out with $\epsilon$ and end with a bound of, say $4 \epsilon$. I disapprove of this, since, in my opinion, introducing magic constants distracts from the following of the proof by the reader, especially the inexperienced reader.