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.

$\frac{x^2}{25}+\frac{y^2}{100}=1$

Answer 1

the equation of given ellipse is
$\frac{x^2}{25}+\frac{y^2}{100}=1$
Now $100>25\Rightarrow a^2=100and{b}^2=25$
So the equation of ellipse in standard form is
$\frac{y^2}{a^2}+\frac{x^2}{b^2}=1\therefore a^2=100\Rightarrow a=10And{b}^2=25\Rightarrow b=5\text{We know that}c=\sqrt{a^2-b^2}\therefore c=\sqrt{100-25}=\sqrt{75}=5\sqrt{3}$
$\therefore \text{Coordinates of foci are}(0,\pm c)i.e.(0,\pm 5\sqrt{3})\text{Coordinates fo vertices are}(0,\pm a)i.e.(0\pm 10)\text{Length of major axis}=2a=2\times 10=20\text{Length of minor axis}=2b=2\times 5=10\text{Eccentricity}\left(e\right)=\frac{c}{a}=\frac{5\sqrt{3}}{10}=\frac{\sqrt{3}}{2}\text{Length of latus rectum}=\frac{2b^2}{a}=\frac{2\times 25}{10}=5$