Originally Posted by

**earboth** Good morning,

1. Let d denotes the distance from C to the directrix. Then you know $\displaystyle d = \frac{25}{21} \cdot \sqrt{21}$

With $\displaystyle a = e \cdot d$ follows: $\displaystyle a = \frac{\sqrt{21}}{5} \cdot \frac{25}{21} \cdot \sqrt{21} = 5$

2. Since $\displaystyle \frac{a^2-b^2}{a^2} = \frac{21}{25}~\implies~b = 2$

3. Therefore the equation of the ellipse is:

$\displaystyle \frac{(x-4)^2}4+\frac{(y+5)^2}{25}=1$

4. The coordinates of the foci are $\displaystyle F_1\left(4\ ,\ -5+\sqrt{21} \right)$ and $\displaystyle F_2\left(4\ ,\ -5-\sqrt{21} \right)$

5. The equation of the upper directrix is $\displaystyle d:y = -5+\frac{25}{21} \cdot \sqrt{21}$

6. The major axis is placed on the straight line a (see attachment). d crosses a in the point $\displaystyle D\left(4, -5+\frac{25}{21} \cdot \sqrt{21} \right)$

7. The midpoint between C and D is $\displaystyle M \left(4, -5+\frac{25}{42} \cdot \sqrt{21} \right)$ which is definitely not the focus F (compare the result at #4)