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I don't always derive the equation of an ellipse but when I do I'm usually given the foci and vertices, and it's usually vertical ;)

View Video Notes: https://steemit.com/mathematics/@mes/video-notes-conic-sections-ellipses-example-2-vertical-ellipse

I can only sigh at the degree of computations here; it is said that numbers and math are but windows to the very fabric of life as God intend it to be. Thanks for these my dear friend.

Ellipses are, by their nature not "perfectly round" in the technical sense that circles are round. And very well explained by video.
Thanks for share ....
keep posting ....