The image of the question if you don't see all the symbols
The given points $p_1,p_2,p_3,p_4$ are located at the vertices of a convex quadrilateral on the real affine plane.
I am looking for an explicit condition on the point $p_5$ necessary and sufficient for the conic which is determined by $p_1,p_2,p_3,p_4,p_5$ to be an ellipse.
Could you give me a hint, please?
I tried to "go" to $\mathbb P_2$ and change the coordinates of these points to more convenient (for example, if $a=(1:a_1,a_2)$ change it to $(1:1:0)$) but I am not sure that this transformation preserves conic
Best Answer
One can always construct two parabolas passing through $p_1$, $p_2$, $p_3$, $p_4$ (green and pink in the figure below), each one possibly degenerating into a couple of parallel lines if two opposite sides of quadrilateral $p_1p_2p_3p_4$ are parallel. Point $p_5$ will determine an ellipse if it lies inside either parabola but not in their intersection.
This follows from the fact that five points always determine a conic section, and because the parabola is a limiting case between ellipse and hyperbola: each time $p_5$ crosses the boundary of a parabola, conic section $p_1p_2p_3p_4p_5$ switches from ellipse to hyperbola (or viceversa).