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

Answer 1

Given equation of ellipse is
$\frac{x^2}{36}+\frac{y^2}{16}=1$
On comparing the given equation with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, we get
$a=6,b=4$
Since, $a>b$ ,therefore the major axis is along the $x$-axis while the minor axis is along the $y$-axis.
Length of major axis $=2a=12$
Length of minor axis $=2b=8$
Eccentricity $e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{\frac{36-16}{36}}=\frac{\sqrt{20}}{6}=\frac{\sqrt{5}}{3}$
Coordinates of the foci are $(2\sqrt{5},0)$ and $(-2\sqrt{5},0)$
The coordinates of the vertices are $(6,0)$ and $(-6,0)$
Length of latus rectum = $\frac{2b^2}{a}=\frac{2\times 16}{6}=\frac{16}{3}$