Question

For the given ellipse, find the length of major and minor axes, coordinates of foci and vertices and the eccentricity.

$3x^2+2y^2=6$

Answer 1

since the given equation is,

$3x^2+2y^2=6$

$\Rightarrow \frac{x^2}{2}+\frac{y^2}{3}=6$

$thisshowsthat,a=\sqrt{2}andb=\sqrt{3}$

substituting these values in the realtion

$a^2=b^2(1-e^2)$

$weget,2=3(1-e^2)$

$\therefore e=\frac{1}{\sqrt{3}}$

$\because a^2<b^2$

therefore major axis is along y-axis and minor axis is along x-axis

therefore length of major axis=2b=$2\sqrt{3}$

and minor axis=2a=$2\sqrt{2}$

therefore the foci are$(0,\pm be)=(0,\pm 1)$

and the vertices are$(0,\pm b)=(0,\pm \sqrt{3})$