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 - 4 \right)^2}{ 9 } + \dfrac{ \left( y - 1 \right)^2}{ 25 } = 1 $ we conclude that:
$$ h = 4, ~~ k = 1, ~~ a^2 = 9 ~~ b^2 = 25 $$ $$ a = \sqrt{ 9 } = 3 ~~\text{and} ~~ b = \sqrt{ 25 } = 5 $$In this example is $ a < b $, so we will use formulas for this case.
Center is $ (h, k) = ( 4, 1 ) $
Major Axis Length is $2 b = 2 \cdot 5 = 10 $
Minor Axis Length is $2 a = 2 \cdot 3 = 6 $
Linear Eccentricity (focal distance) is:
$$ c = \sqrt{b^2 - a^2} = \sqrt{ 25 - 9 } = \sqrt{ 16 } = 4$$Eccentricity is:
$$ e = \dfrac{ c } { a } = \dfrac{ 4 }{ 3 } = \frac{ 4 }{ 3 } $$Area is $ A = a b \pi = 3 \cdot 5 \cdot \pi = 15\pi $
First focus is $ \text{F1} = \left(h, k-c \right) = \left(4, 1 - 4 \right) = \left(1, -3 \right) $
Second focus is $ \text{F2} = \left(h, k + c \right) = \left(4, 1 + 4\right) = \left(1, 5 \right) $
First Vertex is $ \left(h, k - b \right) = \left(4, 1 - 5 \right) = \left(4, -4 \right) $
Second Vertex is $ \left(h, k + b \right) = \left(4, 1 + 5 \right) = \left(4, 6 \right) $
First Co-vertex is $ \left(h - a, k \right) = \left(4 - 3, 1 \right) = \left(1, 1 \right) $
Second Co-vertex is $ \left(h + a, k \right) = \left(4 + 3, 1 \right) = \left(7, 1 \right) $