Question

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

Answer 1

Given: Equation of ellipse is $9x^2+4y^2=36$
i.e.,$\frac{x^2}{4}+\frac{y^2}{9}=1$
Here, $9>4$
So, the major axis is along the $y-\text{axis}$,
while the minor axis is along the $x-\text{axis}$.

On comparing the given equation with $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ (general form), we get

$a^2=9\Rightarrow a=3$
and
$b^2=4\Rightarrow b=2$

Now $c=\sqrt{(a^2-b^2)}=\sqrt{(9-4)}$
$\Rightarrow c=\sqrt{5}$

$a=3,b=2\text{and}c=\sqrt{5}$

$\because$ Major axis is along $y-\text{axis}$, while minor axis is along $x-\text{axis}$.

Then, coordinates of the foci $=(0,\pm c)$
$=(0,\pm \sqrt{5})$

Co-ordinates of the vertices $=(0,\pm a)$
$=(0,\pm 3)$

Length of major axis $=2a=2\left(3\right)=6$
Length of minor axis $=2b=2\left(2\right)=4$
Eccentricity, $e=\frac{c}{a}=\frac{\sqrt{5}}{3}$