Is an ellipse determined by three points and tangent lines at two of them?

Question

I have three points, $A$, $B$, and $C$, on an ellipse; two tangent lines at $A$ and $B$ are also known. Are these enough to determine the whole ellipse?

Answer 1

Answered in comments by OP:

[See] Paris Pamfilos, A Gallery of Conics by Five Elements, Forum Geometricorum, Volume 14 (2014) 295–348. My problem was completely solved

Answer 2

A conic by five distinct points can be constructed as follows: consider the lines $d_{12}=0,d_{34}=0,d_{13}=0$ and $d_{24}=0$ and form the pencil of conics

$$(1-\lambda)d_{12}d_{34}+\lambda d_{13}d_{24}=0.$$ these are all the conics through $1,2,3,4$. Then plug the coordinates of $5$ in the equation and determine $\lambda$ to get the conic by $1,2,3,4,5$.

Now if $1$ and $2$ get closer and closer, $d_{12}$ tends to a tangent, let $t_1$, and the pencil becomes

$$(1-\lambda)t_{1}d_{34}+\lambda d_{13}d_{14}=0.$$

If $4$ also migrates to $3$, defining the tangent $t_3$,

$$(1-\lambda)t_{1}t_3+\lambda d_{13}^2=0.$$