Is it possible to find equation for ellipse when focus, eccentricity and two points are known?

Question

Is it possible to find equation for an ellipse when we know two points and one focus in 2d cartesian coordinate system?

We can also make these assumptions about these two given points depending on the scenario:

Both of these known points are closer to the given focus point instead of the yet unknown focus point or both of these known points are closer to the yet unknown focus point.

If this is possible, how would we do that with this data for example:

$$F_1 = (0,0), P_1 = (3,2), P_2 = (3, -2)$$

Edit:

Seems like it's not possible to get the equation from this information only. What if we know the eccentricity too?

Ie. $e = 0.4$

Answer 1

I think it is not possible to find an unique ellipse. You can choose for example the circle with radius $\sqrt{13}$. The equation for an ellipse is $\frac{(x-x_M)^2}{a^2}+\frac{(y-y_M)^2}{b^2}$. Here $(x_M,y_M)$ is the midpoint of the ellipse. So you have to many degrees of freedom. By symmetrie of the ellipse you know, that the other focus is in $(2x_M,2y_M)$.

Answer 2

If $F_2$ is any point such that $F_1P_1 + F_2P_1 = F_1P_2 + F_2P_2$, then there exists an ellipse through $P_1$ and $P_2$ with foci $F_1$ and $F_2$. $F_2$ satisfies $F_2P_1 -F_2P_2 =$ constant, and the locus of all such points is a hyperbola.

So there is no unique solution.