Question

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 $4x^2+9y^2=36$

Answer 1

The given equation is $4x^2+9y^2=36$
It can be written as
$4x^2+9y^2=36$

$\frac{x^2}{9}+\frac{y^2}{4}=1$
$\frac{x^2}{3^2}+\frac{y^2}{2^2}=1...\left(1\right)$
Here the denominator of $\frac{x^2}{3^2}$ is greater than the denominator of $\frac{y^2}{2^2}$
Therefore, the major axis is along the $x$-axis while the minor axis is along the $y$-axis.
On comparing, the given equation with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, we obtain $a=3$ and $b=2$
$\therefore ae=c=\sqrt{a^2-b^2}=\sqrt{9-4}=\sqrt{5}$
Therefore, the coordinates of the foci are $(\pm \sqrt{5},0)$.
The coordinates of the vertices are $(\pm 3,0)$
Length of major axis $=2a=6$
Length of minor axis $=2b=4$
Eccentricity $e=$ $\frac{c}{a}=\frac{\sqrt{5}}{3}$
Length of latus rectum $=$ $\frac{2b^2}{a}=\frac{2\times 4}{3}=\frac{8}{3}$