Find the coordinates of the foci, the vertices, the lengths of major and minor axes and the eccentricity of the ellipse $9x^2+4y^2=36$.

Solution: Given equation is,
$9x^2+4y^2=36$
We will convert this equation into standard form.
$\Rightarrow \frac{9}{36}{x}^2+\frac{4}{36}{y}^2=1$
$\Rightarrow \frac{x^2}{4}+\frac{y^2}{9}=1$
So, this is our standard equation of ellipse with,
$a=2,b=3$
$c=\sqrt{b^2-a^2}=\sqrt{9-4}=\sqrt{5}$
Here, as $b>a$, major-axis will be $Y-$axis.
Now, foci will be $\left(0,\pm c\right)=\left(0,\pm \sqrt{5}\right).$
Vertices will be $\left(0,\pm b\right)=\left(0,\pm 3\right).$
Length of major-axis $=2b=6$
Length of minor axis $=2a=4$
Eccentricity $=\frac{c}{b}=\frac{\sqrt{5}}{3}.$