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3 added 2 characters in body

It is easy to see that all these parabolas have the same directrix. Height of a derectirix directirix correspond to energy of the body. So you have the family of parabolas with the common point $P$ and the directrix $l$. It is easy to prove, (using just definition of parabola as a locus of points...) that all of the touched the parabola with the focus at $P$ and the directrix $l_1$, which parallel $l$ (actually $l$ is midline of $P$ and $l_1$).

The same holds for sun-earth set. If Earth decide decides to fly in other direction (but with the same speed) its path will be always touch the fixed ellips ellipse with foci in Sun and this position of Earth.

2 added 8 characters in body

It is easy to see that all these parabolas have the same directrix. Height of a derectirix correspond to energy of the body. So you have the family of parabolas with the common point $P$ and the directrix $l$. It is easy to prove, (using just definition of parabola as a locus of points...) that all of the touched the parabola with the focus at $P$ and directris the directrix $l_1$, which parallel $l$ (actually $l$ is midline of $P$ and $l_1$).

The same holds for sun-earth set. If Earth decide to fly in other direction (but with the same speed) its path will be always touch the fixed ellips with foci in Sun and this position of Earth.

1

It is easy to see that all these parabolas have the same directrix. Height of a derectirix correspond to energy of the body. So you have the family of parabolas with the common point $P$ and the directrix $l$. It is easy to prove, (using just definition of parabola as a locus of points...) that all of the touched the parabola with focus at $P$ and directris $l_1$, which parallel $l$ (actually $l$ is midline of $P$ and $l_1$).

The same holds for sun-earth set. If Earth decide to fly in other direction (but with the same speed) its path will be always touch the fixed ellips with foci in Sun and this position of Earth.