Question

In each of the Exercises $1$ to$9$, 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 $\frac{x^2}{4}+\frac{y^2}{25}=1----\left(1\right)$

$\to$ Length of major axis $=2b=10$

$\to$ length of minor axis $=2a=4$

$\to$ eccentricity $e=\sqrt{\frac{b^2-a^2}{b^2}}=\sqrt{1-\frac{a^2}{b^2}}=\frac{\sqrt{21}}{3}$

$\to$ Centre $=(0,0)$

$\to$ vertices $\to (0,5)$ & $(0,-5)$

$\to$ Co-verticed $=(2,0)$ & $(-2,0)$

$\to$ focii $=(0,\sqrt{21})$ & $(0,-\sqrt{21})$

$\to$ length of latus rectum $=\frac{2a^2}{b}=\frac{2\times 4}{5}=\frac{8}{5}$