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 - 0 \right)^2}{ \frac{ 406 }{ 125 } } + \dfrac{ \left( y - 0 \right)^2}{ \frac{ 463 }{ 200 } } = 1 $ we conclude that:
$$ h = 0, ~~ k = 0, ~~ a^2 = \frac{ 406 }{ 125 } ~~ b^2 = \frac{ 463 }{ 200 } $$ $$ a = \sqrt{ \frac{ 406 }{ 125 } } = \frac{\sqrt{ 2030 }}{ 25 } ~~\text{and} ~~ b = \sqrt{ \frac{ 463 }{ 200 } } = \frac{\sqrt{ 926 }}{ 20 } $$In this example is $ a > b $, so we will use formulas for this case.
Center is $ (h, k) = ( 0, 0 ) $
Major Axis Length is $2 a = 2 \cdot \frac{\sqrt{ 2030 }}{ 25 } = \frac{ 2 \sqrt{ 2030}}{ 25 } $
Minor Axis Length is $2 b = 2 \cdot \frac{\sqrt{ 926 }}{ 20 } = \frac{\sqrt{ 926 }}{ 10 } $
Linear Eccentricity (focal distance) is:
$$ c = \sqrt{a^2 - b^2} = \sqrt{ \frac{ 406 }{ 125 } - \frac{ 463 }{ 200 } } = \sqrt{ \frac{ 933 }{ 1000 } } = \frac{\sqrt{ 9330 }}{ 100 }$$Eccentricity is:
$$ e = \dfrac{ c } { a } = \dfrac{ \frac{\sqrt{ 9330 }}{ 100 } }{ \frac{\sqrt{ 2030 }}{ 25 } } = \frac{\sqrt{ 189399 }}{ 812 } $$Area is $ A = a b \pi = \frac{\sqrt{ 2030 }}{ 25 } \cdot \frac{\sqrt{ 926 }}{ 20 } \cdot \pi = \frac{\sqrt{ 469945 }}{ 250 }\pi $
First focus is $ \text{F1} = \left(h - c, k \right) = \left(0 - \frac{\sqrt{ 9330 }}{ 100 }, 0 \right) = \left(- \frac{\sqrt{ 9330 }}{ 100 }, 0 \right) $
Second focus is $ \text{F2} =\left(h + c, k \right) = \left(0 + \frac{\sqrt{ 9330 }}{ 100 }, 0 \right) = \left(\frac{\sqrt{ 9330 }}{ 100 }, 0 \right) $
First Vertex is $ \left(h - a, k \right) = \left(0 - \frac{\sqrt{ 2030 }}{ 25 }, 0 \right) = \left(- \frac{\sqrt{ 2030 }}{ 25 }, 0 \right) $
Second Vertex is $ \left(h + a, k \right) = \left(0 + \frac{\sqrt{ 2030 }}{ 25 }, 0 \right) = \left(\frac{\sqrt{ 2030 }}{ 25 }, 0 \right) $
First Co-vertex is $ \left(h, k - b \right) = \left(0, 0 - \frac{\sqrt{ 926 }}{ 20 } \right) = \left(0, - \frac{\sqrt{ 926 }}{ 20 } \right) $
Second Co-vertex is $ \left(h, k + b \right) = \left(0, 0 + \frac{\sqrt{ 926 }}{ 20 } \right) = \left(0, \frac{\sqrt{ 926 }}{ 20 } \right) $