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.
$\frac{x^2}{36}+\frac{y^2}{16}=1$

Answer 1

Given $\frac{x^2}{36}+\frac{y^2}{16}=1---\left(1\right)$

Equation $\left(1\right)$ is ellipse where $[a=6,b=4]$

eccentricity $e=\frac{c}{a}=\sqrt{\frac{a^2-b^2}{a^2}}=\sqrt{\frac{1-b^2}{a^2}}=\sqrt{\frac{1-16}{36}}=\frac{\sqrt{20}}{6}$

$[e=\frac{\sqrt{5}}{3}]$

$\to$ Centre is all $(0,0)$ or $(h,k)$

$\to$ Vertices is at $(a,0),(-a,0)=(6,0)$ & $(-6,0)$

$\to$Co-vertices $(0,b)$ & $(0,-b)=(0,4)$ & $(0,4)$ & $(0,-4)$

$\to$ length of mi major axis $=a=6;$ length of major axis $=12$

$\to$ length of mi minor axis $=b=4;$ length of minor axis $=8$

$\to$ focii $=(h+c,k),(h-c,k)=(\sqrt{20},0)$ & $(-\sqrt{20},0)$

$\to$ length of latus rectum $=\frac{2b^2}{a}=\frac{2\times 16}{3}=\frac{16}{3}$