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 $\frac{x^2}{4}+\frac{y^2}{25}=1$

Answer 1

Given equation of ellipse is
$\frac{x^2}{4}+\frac{y^2}{25}=1$
On comparing the given equation with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, we get
$a=2,b=5$
Since, $b>a$ ,therefore the major axis is along the $y$-axis while the minor axis is along the $x$-axis.
Length of major axis $=2b=10$
Length of minor axis $=2a=4$
Eccentricity $e=\sqrt{1-\frac{a^2}{b^2}}=\sqrt{\frac{25-4}{25}}=\frac{\sqrt{21}}{5}$
Coordinates of the foci are $(0,\sqrt{21})$ and $(0,-\sqrt{21})$
The coordinates of the vertices are $(0,5)$ and $(0,-5)$
Length of latus rectum = $\frac{2a^2}{b}=\frac{2\times 4}{5}=\frac{8}{5}$