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**Altmer** Show that the centre C of the ellipse $\displaystyle r = \frac{ep}{1 + ecos(\theta)} $ has the coordinates $\displaystyle \lgroup \frac{e^2p}{1 - e^2}, -\pi \rgroup $

Where e is the eccentricity and p the distance from focus to directrix.

Here is my attempt

The vertices of the major axis occur at 0 and $\displaystyle \pi $

The vertices of the minor axis occur at $\displaystyle \frac{\pi}{2} $ and $\displaystyle \frac{-\pi}{2} $

First coordinate by midpoint rule $\displaystyle \frac{\frac{ep}{1 - e} + \frac{ep}{1 + e}}{2} = \frac{\frac{2ep}{1 - e^2}}{2} = \frac{ep}{1 - e^2} $

By similar reasoning, second coordinate by midpoint rule yields ep.

So I get $\displaystyle C\lgroup \frac{ep}{1 - e^2}, ep \rgroup $

Where have I gone wrong?