Question

Find the length of major axis, the eccentricity the latus rectum, the coordinate of the centre, the foci, the vertices and the equation of the directrices of following ellipse:
$\frac{x^2}{16}+\frac{y^2}{9}=1.$

Answer 1

Given equation of ellipse is
$\frac{x^2}{16}+\frac{y^2}{9}=1$
On comparing the given equation with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, we get
$a=4,b=3$
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 $=2a=8$
Length of minor axis $=2b=6$
Eccentricity $e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{\frac{16-9}{16}}==\frac{\sqrt{7}}{4}$
Coordinates of the foci are $(\sqrt{7},0)$ and $(-\sqrt{7},0)$
The coordinates of the vertices are $(4,0)$ and $(-4,0)$
Length of latus rectum = $\frac{2b^2}{a}=\frac{2\times 9}{4}=\frac{9}{2}$