for ellipse use this sub
and do same sub for the second question
I want to get the polar equation for a ellipse in polar form with the center at one of the foci. I started with the equation of an ellipse in Cartesian coordinates:
Using and I obtained:
Now I would like the same thing but centered at a focus:
From wiki, I found that the distance between the foci is . I not sure why that is the case and it would be nice to see some proof. Then I reasoned the ellipse must be shifted half that distance as below:
Using the identity I obtained:
I'm kindda stuck here. I know that I should get something like
Where will the term came from? and the ? Am I on the right track?
Am I doing something wrong?
I am actually using this in a physics problem showing that the orbital velocity at the point which the minor axis meets the elliptical orbit is the geometrical mean of the velocities at the ends of the major axis. With this identity:
my proof works very well, but I don't want to just pull an equation off wiki and rest my proof on that. I would like to derive that equation by doing a shift of coordinates.