Ellipse centered at a focus

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

Therefore:

Where will the term came from? and the ? Am I on the right track?

Am I doing something wrong?