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 equation of given ellipse is,
$4x^2+9y^2=36$
$i.e.\frac{4x^2}{36}+\frac{9y^2}{36}=1\Rightarrow \frac{x^2}{9}+\frac{y^2}{4}=1\text{Now}9>4\Rightarrow a^2=9and{b}^2=4\text{So the equation of ellipse in standard form is},\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\therefore a^2=9\Rightarrow a=3and{b}^2=4\Rightarrow b=2\text{We know that}c=\sqrt{a^2-b^2}\therefore c=\sqrt{9-4}=\sqrt{5}\text{Coordinates of foci are}(\pm c,0)i.e.(\pm \sqrt{5},0)\text{Coordinates of vertices are}(\pm a,0)i.e.(\pm 3,0)\text{Length of major axis}=2a=2\times 3=6\text{Length of minor axis}=2b=2\times 2=4\text{Eccentricity}\left(e\right)=\frac{c}{a}=\frac{\sqrt{5}}{3}\text{Length of latus rectum}=\frac{2b^2}{a}=\frac{2\times 4}{3}=\frac{8}{3}$