This is a rough sketch to roughly explain the terms I will be using:
Let's say you make a machine that would rotate the paper in some frequency $x$, let's say 1 rotation per second.
The drawing mechanism is made out of one leg that's fixed in the paper at the middle, and one that's sticking away by some distance. In state a, that distance is $a$, and in state b, that distance is $b$. $a$ and $b$ will also be the axii of the ellipse that will hopefully come into existence.
If only the paper is rotating, that would obviously make a circle, since the distance would be constant.
What if the machine also moves the radius from a to b periodically, with a frequency of $2x$. Kinda like a pendulum, where one extreme point is radius a and the other is radius b. This pendulum has a period that's one half of the period of the spinning paper.
It makes some sense. If the ellipse center is in $(0,0)$, and we start this machine so the pencil part of the drawing thing is on $(0,a)$ it will also draw $(b,0), (0,-a), (-b,0)$. Those are also points of the ellipse with axii $a$ and $b$. I have no idea how to check if it would work for other points too.
Well you are describing a curve with polar equation:
where $a,b,k$ are some constants, $t$ is the angle from some axis (usually $x$-axis) and $r$ is the distance from origin.
In your question you chose $k = 2$, but that does not give you an ellipse. For example if you pick $(a,b) = (2,5)$ you get a peanut shape. You can use Graph or Geogebra to plot these and see for yourself.
Wikipedia gives you the polar equation of an ellipse. If you plot $r$ against $t$ it will give you quite a different curve from the path traced by a pendulum.