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.
$\frac{x^2}{100}+\frac{y^2}{400}=1.$

Answer 1

The equation of give ellipse is,
$\frac{x^2}{100}+\frac{y^2}{400}=1.$
Now $400>100\Rightarrow a^2=400and{b}^2=100$
so the equation of ellipse in standard form is,
$\frac{y^2}{a^2}+\frac{x^2}{b^2}=1.\therefore a^2=400\Rightarrow a=20\text{and}{b}^2=100\Rightarrow b=10\text{We know that}c=\sqrt{a^2-b^2}\therefore c=\sqrt{400-100}=\sqrt{300}=10\sqrt{3}\text{Coordinates of foci are}(0,\pm c)i.e.(0,\pm 10\sqrt{3})\text{Coordinates of vertices are}(0,\pm a)i.e.(0,\pm 20)\text{Length of major axis}=2a=2\times 20=40\text{Length of minor axis}=2b=2\times 10=20\text{Eccentricity}\left(e\right)=\frac{c}{a}=\frac{10\sqrt{3}}{20}=\frac{3}{2}\text{Length of latus rectum}=\frac{2b^2}{a}=\frac{2\times 100}{20}=10.$