The standard equation of an ellipse is $ \dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1 $. Comparing to the equation of our ellipse $ \dfrac{ \left( x - 18 \right)^2}{ 400 } + \dfrac{ \left( y + 12 \right)^2}{ \frac{ 78751 }{ 200 } } = 1 $ we conclude that:
$$ h = 18, ~~ k = -12, ~~ a^2 = 400 ~~ b^2 = \frac{ 78751 }{ 200 } $$ $$ a = \sqrt{ 400 } = 20 ~~\text{and} ~~ b = \sqrt{ \frac{ 78751 }{ 200 } } = \frac{\sqrt{ 157502 }}{ 20 } $$In this example is $ a > b $, so we will use formulas for this case.
Center is $ (h, k) = ( 18, -12 ) $
Major Axis Length is $2 a = 2 \cdot 20 = 40 $
Minor Axis Length is $2 b = 2 \cdot \frac{\sqrt{ 157502 }}{ 20 } = \frac{\sqrt{ 157502 }}{ 10 } $
Linear Eccentricity (focal distance) is:
$$ c = \sqrt{a^2 - b^2} = \sqrt{ 400 - \frac{ 78751 }{ 200 } } = \sqrt{ \frac{ 1249 }{ 200 } } = \frac{\sqrt{ 2498 }}{ 20 }$$Eccentricity is:
$$ e = \dfrac{ c } { a } = \dfrac{ \frac{\sqrt{ 2498 }}{ 20 } }{ 20 } = \frac{\sqrt{ 2498 }}{ 400 } $$Area is $ A = a b \pi = 20 \cdot \frac{\sqrt{ 157502 }}{ 20 } \cdot \pi = \sqrt{ 157502 }\pi $
First focus is $ \text{F1} = \left(h - c, k \right) = \left(18 - \frac{\sqrt{ 2498 }}{ 20 }, -12 \right) = \left(15.501, -12 \right) $
Second focus is $ \text{F2} =\left(h + c, k \right) = \left(18 + \frac{\sqrt{ 2498 }}{ 20 }, -12 \right) = \left(20.499, -12 \right) $
First Vertex is $ \left(h - a, k \right) = \left(18 - 20, -12 \right) = \left(-2, -12 \right) $
Second Vertex is $ \left(h + a, k \right) = \left(18 + 20, -12 \right) = \left(38, -12 \right) $
First Co-vertex is $ \left(h, k - b \right) = \left(18, -12 - \frac{\sqrt{ 157502 }}{ 20 } \right) = \left(18, -31.8433 \right) $
Second Co-vertex is $ \left(h, k + b \right) = \left(18, -12 + \frac{\sqrt{ 157502 }}{ 20 } \right) = \left(18, 7.8433 \right) $