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}{ 12 } + \dfrac{ \left( y - 0 \right)^2}{ 26 } = 1 $ we conclude that:
$$ h = 0, ~~ k = 0, ~~ a^2 = 12 ~~ b^2 = 26 $$ $$ a = \sqrt{ 12 } = 2 \sqrt{ 3 } ~~\text{and} ~~ b = \sqrt{ 26 } $$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 b = 2 \cdot \sqrt{ 26 } = 2 \sqrt{ 26 } $
Minor Axis Length is $2 a = 2 \cdot 2 \sqrt{ 3 } = 4 \sqrt{ 3 } $
Linear Eccentricity (focal distance) is:
$$ c = \sqrt{b^2 - a^2} = \sqrt{ 26 - 12 } = \sqrt{ 14 } $$Eccentricity is:
$$ e = \dfrac{ c } { a } = \dfrac{ \sqrt{ 14 } }{ 2 \sqrt{ 3 } } = \frac{\sqrt{ 42 }}{ 6 } $$Area is $ A = a b \pi = 2 \sqrt{ 3 } \cdot \sqrt{ 26 } \cdot \pi = 2 \sqrt{ 78 }\pi $
First focus is $ \text{F1} = \left(h, k-c \right) = \left(0, 0 - \sqrt{ 14 } \right) = \left(0, -\sqrt{ 14 } \right) $
Second focus is $ \text{F2} = \left(h, k + c \right) = \left(0, 0 + \sqrt{ 14 }\right) = \left(0, \sqrt{ 14 } \right) $
First Vertex is $ \left(h, k - b \right) = \left(0, 0 - \sqrt{ 26 } \right) = \left(0, -\sqrt{ 26 } \right) $
Second Vertex is $ \left(h, k + b \right) = \left(0, 0 + \sqrt{ 26 } \right) = \left(0, \sqrt{ 26 } \right) $
First Co-vertex is $ \left(h - a, k \right) = \left(0 - 2 \sqrt{ 3 }, 0 \right) = \left(-2 \sqrt{ 3 }, 0 \right) $
Second Co-vertex is $ \left(h + a, k \right) = \left(0 + 2 \sqrt{ 3 }, 0 \right) = \left(2 \sqrt{ 3 }, 0 \right) $