Let $A(0,1)$, $B(0,-1)$, and $C_t(x,y)$ such that $AC_t = t$ and $BC_t = t + 1$. Another way to say it is that the triangle $A\,B\,C_t$ has one fixed (vertical) side of length $2$, one side of length $t$ and one of length $t+1$.
Question: How to find a description of the path of $C_t$ when we make $t$ vary in $\mathbb{R}^+$?
I'm looking for a description as $y = f(x)$, or if not possible, the addition of two (or more) curves $y=f(x)$.
Note:
this comes from a triangulation problem (time difference between sound events and GPS coordinates, I will link a complete description when finished), see "Application" in this question.
I've tried it by drawing lots of circles on a sheet of paper, the path of $C_t$ seems to be a line, but not 100% sure
Analytically, I found something like $y = f(x) = \frac{2}{\sqrt{3}} x + \frac 12$, but the computations were horrible, not elegant (there's probably much clever), and there are probably other possible paths $y = g(x)$ with a different $g$
The path is a branch of the hyperbola having $A$ and $B$ as foci. Because that hyperbola is the locus of points $C$ such that $|CB-CA|$ is constant.
In your case focal distance is $c=1$ while semi-major axis is $a=|CB-CA|/2=1/2$. Semi-conjugate axis is then $b=\sqrt{c^2-a^2}=\sqrt3/2$ and the equation of the hyperbola is $$ 4y^2-{4\over3}x^2=1. $$