Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.$4x^2+9y^2=36$

Solution: Given equation is,
$4x^2+9y^2=36$
We will convert this equation into standard form.
$\Rightarrow \frac{4}{36}{x}^2+\frac{9}{36}{y}^2=1$
$\Rightarrow \frac{x^2}{9}+\frac{y^2}{4}=1$
So, this is our standard equation of ellipse with,
$a=3,b=2$
$c=\sqrt{a^2-b^2}=\sqrt{9-4}=\sqrt{5}$
Here, as $a>b$, major-axis will be $X-$axis.
Now, foci will be $\left(\pm c,0\right)=\left(\pm \sqrt{5},0\right).$
Vertices will be $\left(\pm a,0\right)=\left(\pm 3,0\right).$
Length of major-axis $=2a=6$
Length of minor axis $=2b=4$
Eccentricity $=\frac{c}{a}=\frac{\sqrt{5}}{3}.$
Length of latus rectum $=2\frac{b^2}{a}=2\ast \frac{4}{3}=\frac{8}{3}.$