what is mathematical description of parabola produced by intersection of two evolving circles? this video makes one wonder about what is mathematical description of parabola produced by intersection of two evolving circles?
is this description having a temporal dimension makes it relavant physics and not math?
To help clarify the question a bit, having looked at the video, let me rephrase it, in a way that the OP will, I hope, approve:
Given two distinct points $P$ and $Q$, you can build a circle $C_t$ of radius $t$ about $P$ and a circle of $C'_t$ of radius $t + a$ about $Q$ for some constant $a$. For large enough values of $t$, these intersect, and if we consider such intersections, for all $t$, they seem to all lie on one hyperbola. Can you mathematically show that this collection of intersections forms a hyperbola, and describe the hyperbola in terms of $P, Q,$ and $a$? 
 A: To answer my own rephrasing of the question: 
Suppose that $X$ is a point of $C_t \cap C'_t$. Then because it's on $C_t$, the distance from $X$ to $P$ is $t$ --- the radius of $C_t$ is $t$, after all. Similarly, the distance from $X$ to $Q$ is $t + a$. The difference between these two distances is $(t+a) - t = a$, i.e., it's independent of $t$. 
Now suppose instead that $X$ is a point whose distance to $Q$ is $a$ greater than its distance to $Q$. If we let $t$ be the distance from $X$ to $P$, then we see that $X$ lies on $C_t$, and similarly that it lies on $C'_t$. 
In summary: the points of $C_t \cap C'_t$, for all possible values of $t$, are exactly the points for which the distance to $Q$ is $a$ greater than the distance to $P$. 
Since a hyperbola is typically described by saying that it's the set of points where the difference of the distances to two points is a constant, you can see that the set I've described is exactly one part of a hyperbola. (The other "arc" consists of points where the distance to $P$ minus the distance to $Q$ is $a$.)
So: what looks like a hyperbola in the video really is a hyperbola, and it's the one defined by 
$$
\{ x \mid d(X, Q) - d(X, P) = a \}
$$
where $a$ is the size of the circle around $Q$ at the time that the circle around $P$ starts growing. 
