I feel like such an ellipse is not unique but I don't have the mathematical background to prove it, can you help me ?
The context is a physical problem which can be reduced to : Find the eccentricity of an ellipse with a given tilted angle only measuring one of its point. Would measuring two points be sufficient ?
Thanks,
Anatole Jimenez
You can bring the main axis on the horizontal axis by a $-\theta$ rotation, with point $(x_p,y_p)$ transformed into:
$$\begin{cases}x'_p&=& \ \ \ \cos \theta \ x &+& \sin \theta \ y\\ y'_p&=&-\sin \theta \ x &+&\cos \theta \ y\end{cases}$$
Therefore you have to find $a$ and $b$ such that:
$$\frac{(x'_p)^2}{a^2}+\frac{(y'_p)^2}{b^2}=1$$
One equation with 2 unknowns indeed isn't enough.
You need a second point in order to have a second equation.
Setting $A=1/a^2$ and $B=1/b^2$, you will get a linear system which will have in general solutions (unless its determinant is zero).
But it will not work always because these solutions $A,B$ must be positive !
Maybe this last issue will not be bothering if your points have a physical meaning.