Question

Find the length of the latus rectum, the eccentricity and the coordinates of the foci of the ellipse
$x^2+3y^2=a^2$

Answer 1

$\frac{x^2}{a^2}+\frac{y^2}{a^2/3}=1$

$A=\pm aB=\pm a/\sqrt{3}$

$e=\sqrt{1-\frac{\left(a^2/3\right)}{\left(a^2\right)}}=\sqrt{\frac{2}{3}}a$

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

$=(0,\pm \frac{a^2}{3}\sqrt{2})$

Length of $LR=\frac{2a^2}{b}=\frac{2\left(a^2\right)}{a/\sqrt{3}}=\frac{2a}{\sqrt{3}}$