Here's what I know:
- The coordinates of a point A on an ellipse
- The instantaneous slope of the ellipse at point A
- Coordinates of one focus of the ellipse
Here's what I'm trying to find:
- The equation of the ellipse (ideally parametric, but converting is easy)
The ellipse may or may not be at the center, and may or may not be rotated.
I want to know if this is possible - does the provided information define only one ellipse - and if the information defines only a few ellipses (not an infinite number) how could I get their equations from the known information.
What I've tried: I took the derivative of the equation of an ellipse, set it equal to the slope, plugged in the x and y of the point, and plugged in the focal length (substituting $a^2$ for $b^2+c^2$).
This works when the ellipse is not rotated and is at the origin, since the only unknown is $b$. After solving for $b$, you can plug it back in to the original equation and solve for $a$. When the ellipse is rotated and/or shifted from the origin, there are too many unknowns, though...
(This is my first post here, so let me know if I've made any mistakes!)
These conditions do not define a unique ellipse [many thanks to @Semiclassical for pointing out an error in a prior answer].
In fact, due to the "reflection (or caustic) property" of the ellipse, the "ray" emitted from focus $F$ to the given point, say $M_0$, is reflected on the given tangent, and the 2nd focus $F'$ can be any point on this half line. Once you know the two foci of an ellipse and a point on it, the definition of an ellipse (set of points $M$ such that $MF+MF'=$ constant) gives a unique ellipse as solution.