Finding formula for set of ellipses

Question

I'm looking for a formula for a set of ellipses lying on the intersections of two set of circles. The python code for the two sets of circles is as follows:

for x in range(0,180):
  circles = circles + '<circle cx="-30" cy="-50" r="' + str(x*5) +  '" stroke="black" fill="none" stroke-width="1" />\n'

for x in range(0,180):
  circles = circles + '<circle cx="510" cy="-50" r="' + str(x*5) +  '" stroke="black" fill="none" stroke-width="1" />\n'

I made an image of the two sets of circles, and added two examples of the ellipses (red) I'm looking for.

figure

I have no idea how to approach the formula and therefor very grateful for any hint and answere!

Answer 1

From the code and the figure, it is evident that you are looking for ellipses whose distance from the two points $(-30, -50)$ and $(510, -50)$ add up to a constant. Those are precisely the ellipses whose foci are those two points. These ellipses have equations of the form

$$ \left(\frac{x-240}{a}\right)^2+\left(\frac{y-(-50)}{b}\right)^2 = 1 $$

where the center of the ellipse is at $(240, -50)$, the semimajor axis $a > 510-240 = 240-(-30) = 270$ (note the strict inequality), and $a^2-b^2 = 270^2$.