The included drawing shows what I am interested in determining. I'm trying to get the equation for the points on the ellipse when the origin is at P. P should not be assumed to be a focus of the ellipse. So, I'm not sure that using equations for an ellipse where the center is shifted away from $0,0$ apply. This isn't about shifting an origin which is the center. Its about changing the origin from the center to another point along the x axis.
