I have this in mind since quite a time. In book it was given that once kepler asked newton what is relation of distance with gravitational force. Newton said inverse square on which kepler concluded motion of planets will be elliptical, it is more easy to generalize it to circle as older attempts. I understand that circle is special eclipse with $a=b$.
So my question is how inverse square determines/ derives to elliptical path motion?
Johannes Kepler died in 1630 and Isaac Newton was not born until 1642, so they never met. Kepler had determined from astronomical observations that the planets known at that time followed orbits that were ellipses with the sun at one focus. It was Edmond Halley who had the conversation about Kepler's laws with Newton in 1684. This then led to Newton writing and publishing the Principia Mathematica in which he showed that Kepler's laws are a consequence of his law of universal gravitation.
Determining the possible shapes of orbits for two bodies subject to an inverse square law of attraction is known as the Kepler problem. The possible orbits turn out to be conic sections, which includes circles and ellipses (closed orbits) as well as parabolas and hyperbolas (open orbits).
Well, it basically follows from the direct application of 2nd Newton's Law for the body. This was originally elaborated within a geometrical approach in his famous Principia (1687). See, for example, the development available in this paper: https://arxiv.org/pdf/1805.08872.pdf
I believe in Goldstein you can also find valuable information on the chapter dedicated to central forces. Once you find Eq. 4 in the paper mentioned above, using polar coordinates, you solved your problem.
Basically you have a two-dimensional problem to solve in radial e angular coordinates, $r$ and $\theta$. One can reduce the problem to just one radial coordinate using the conservation of angular momentum $L$.
Then you have a equation in $r$ $$\frac{d^2 r}{dt^2}= - \frac{GM}{r^2}+\frac{L^2}{mr^3}.$$
It is useful to perform a change of variable $u=1/r$ to solve this differential equation, and demonstrate that
$$u= \frac{1}{r}= \frac{GMm^2}{L^2} + A cos \theta ,$$
which can be rewritten as the angular equation of the elipse
$$\frac{a(1-\epsilon ^2)}{r}= 1+ \epsilon \cos \theta$$.
See more details in the reference: http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/KeplersLaws.htm#:~:text=We%20now%20back%20up%20to,in%20a%20radial%20inward%20direction.
PS.: This reference contains a demonstration of the first and third Kepler's laws. The second follows directly from the conservation of angular momentum.
