Many detailed proofs are available for Kepler's first law, like this [one](https://physics.stackexchange.com/questions/112668/derivation-of-keplers-laws). But all of them use polar co-ordinates. There is [one](https://physics.stackexchange.com/questions/86435/proving-keplers-1st-law-without-differential-equations) which doesn't even use differential equations. I'm looking for a proof which is done using **cartesian co-ordinates**.

For simplicity lets assume the Sun is at origin and the planet is located on the x-axis and is having a suitable velocity which is parallel to positive y-axis.

Background: I'm a high school physics teacher. My students are not familiar with polar co-ordinates, however they are comfortable with calculus