How was the discovery of the ellipses orbit made?

Answer 1

It was the result of hard work by Johannes Kepler, working at the start of the 17th century on new observations by Tycho Brahe.

Kepler had a set of measurements of the planets' positions, and over a period of months and years he tried to explain the movements of the planets, because they did not exactly fit the old theories of Ptolemy and Copernicus.

So he said to himself, let's try an ellipse, and it worked. At that stage he did not have a reason for the elliptical orbit, other than that it fitted the observations. Much later, Newton explained how a planet moving in the Sun's gravity must follow an elliptical orbit.