Question

Find the length of the latus rectum, the eccentricity and the coordinates of the foci of the ellipse
$5x^2+4y^2=1$

Answer 1

$5x^2+4y^2=1$

$\frac{x^2}{4}+\frac{y^2}{5}=\frac{1}{20}$ or $\frac{x^2}{1/5}+\frac{y^2}{1/4}=1$

$a^2=1/5b^2=1/4$

$a=1/\sqrt{5}b=\frac{1}{2}$

$e=\sqrt{1-\frac{a^2}{b^2}}\Rightarrow e=\frac{1}{\sqrt{5}}$

coordinate of foci $=(0,\pm be)$

$=(0,\frac{\pm 2}{2\sqrt{5}})$

Length of Latus rectum $=\frac{2a^2}{b}=\frac{4}{5}$