We can show that when a particle is projected from a certain height (from the surface of the Earth) with a speed lesser than the orbital speed and in a direction tangential to the surface of the Earth at that point, it follows an elliptical path with the center of the Earth as a focus. The process of deriving this result does not involve putting any lower bound on the speed of projection. Certainly, if the speed of projection is small enough then the trajectory would actually intersect the Earth but we can nevertheless consider the section of the trajectory until it actually hits the Earth to be elliptical. However, when we explicitly consider only those situations when the trajectory of the projectile does intersect the Earth, via approximations , we can show that if the height is sufficiently smaller (than the radius of the Earth) then the trajectory will actually be a parabola. In none of the two calculations, we have assumed the earth to be a point or a flat surface. So is it true that a very small portion of an ellipse is parabola even if both are entirely different conics by definition?
: Via "approximations", I meant treating an approximated gravitational field and solving for the trajectory anew rather than incorporating the approximation into the already evaluated original trajectory. Of course, it would give the same answer (as it should), but unaware of how to Taylor expand a curve, when I posted the question, I saw in this result a paradox as to how could two different conics be the same in some regime.