Question

Find the lengths of the major and minor axes, coordinates of the vertices and the foci, the eccentricity and length of the latus rectum of the ellipse $4x^2+9y^2=144.$

Answer 1

Given equation of ellipse is $4x^2+9y^2=144$

i.e. $\frac{x^2}{36}+\frac{y^2}{16}=1$

On comparing $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ we get

$a^2=36$ and $b^2=16\Rightarrow a=6$ and $b=4$

Here, we see that $a^2>b^2$

$\therefore$ Foci lie on x-axis.

Now.. $c=\sqrt{a^2-b^2}=\sqrt{36-16}=\sqrt{20}=2\sqrt{5}$

Length of the major axis = $2a=2\times 6=12$

Length of the minor axis = $2b=2\times 4=8$

Coordinates of vertices = $(\pm a,0)=(\pm 6,0)$

Coordinates of foci = $(\pm c,0)=(\pm 2\sqrt{5},0)$

Eccentricity of ellipse = $\frac{c}{a}=\frac{2\sqrt{5}}{6}=\frac{\sqrt{5}}{3}$

Length of latus rectum = $\frac{2b^2}{a}=\frac{2\times 16}{6}=\frac{16}{3}$ units