I have a situation for which I have made a very crude drawing. Let's say we have an ellipse in $\mathbb{R}^2$ that is fixed at $x_0 = -a$ and $x_1 = a$ (as if it were resting on two poles). I am curious to know how we can discuss the relative motion of the curve if we were to move one of the "poles" away such that $x_1^{'} = x_1 + \Delta x$ for some $\Delta x \in \mathbb{R}$.
Any assistance would be greatly appreciated.
This is an outline of a solution to your problem. I’m going to rename some of your quantities to conform to conventions about equations of an ellipse. We have an ellipse $(x/a)^2+(y/b)^2=1$, the distance $d$ of the pivot point $P$ from the $y$-axis and the moving point $Q$. The coordinates of these points are $$P = \left(-d,-\frac ba\sqrt{a^2-d^2}\right) \\ Q=\left(d+\Delta,\frac ba\sqrt{a^2-d^2}\right) = (-x_P+\Delta, y_P).$$ As $Q$ moves horizontally, the ellipse pivots around $P$ to maintain contact with $Q$. The point of contact on the unrotated ellipse satisfies $PR=PQ$, i.e., it is the other end of a chord from $P$ with length $PQ$. This suggests that $R$ can be found by computing the intersection of a circle with the ellipse. We’ll come back to this later. Since the segment $\overline{PQ}$ is horizontal, it’s fairly easy to work out that the rotation angle $\theta$ satisfies $$\cos\theta = {x_R-x_P\over x_Q-x_P}, \sin\theta = -{y_R-y_P\over x_Q-x_P}, \tan\theta = -{y_R-y_P\over x_R-x_P}.$$ The center of the rotated ellipse can be found in various ways to be $$x_c = x_P-x_P\cos\theta+y_P\sin\theta \\ y_c = y_P-y_P\cos\theta-x_P\sin\theta$$ and an equation of the rotated ellipse is therefore $${(x\cos\theta-y\sin\theta-x_c)^2\over a^2}+{(x\sin\theta+y\cos\theta-y_c)^2\over b^2}=1.$$
All that’s left is to compute point $R$ and then do some tedious algebra to express the above in terms of $d$ and $\Delta$. The general solution to the system $${x^2\over a^2}+{y^2\over b^2}=1 \\ (x+d)^2+\left(y+\frac ba\sqrt{a^2-d^2}\right)^2=(2d+\Delta)^2$$ is quite messy, though, and there’s still the matter of selecting the correct solution. It might be worth exploring parameterizing by the distance between $P$ and $Q$ instead of the the offset $\Delta$ from the original position of $Q$ to see if it simplifies any of the resulting expressions.