If you assume the location of the first focus to be at the coordinates $(0,0)$ there are still an infinite amount of different conic sections that passes through two arbitrary points (in a $2D$ coordinate system)
This is because there are an infinite amount of locations the second focus can be as show by this calculator here : (https://www.geogebra.org/calculator/fungwd68)
If you move the location of the second focus around the focus-hyperbola (which is the path where the second focus could lie to satisfy the conditions of passing through the two arbitrary points I mentioned above) you will find a location where the semi-major axis is the lowest (which seems to be the turning point of the focus-hyperbola).
Though the conic would obviously be a hyperbola with an eccentricity slightly above $1$ (since the length of their semi-major axis is negative), but I want it to be an ellipse since I plan to use this to model gravity (where an ellipse has a lower energy than a hyperbola).
Because I could not fit all of my question in the title I will finalise it here:
For a conic section (that has an eccentricity less than 1) which the location of the first focus is know ($F_1$ at location $(0,0)$) that lies on two arbitrary points $P_1,P_2$. What would be the equation of the conic and the location of the second focus ($F_2$) where the semi-major axis has the lowest possible value?
The length of the major axis of the ellipse is $P_1F_0+P_1F_1$ (or $P_2F_0+P_2F_1$, which is the same if the conic is an ellipse). But $P_1F_0$ is fixed, hence the major axis attains its minimum value when the distance $P_1F_1$ (or $P_2F_1$) is minimum. But $F_1$ lies on a hyperbola with foci $P_1$ and $P_2$, hence we get the minimum when $F_1$ lies at one of the vertices of that hyperbola.