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 $4 x ^{2}+9 y ^{2}=36$
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 $4 x ^{2}+9 y ^{2}=36$
The given equation is $4 x ^{2}+9 y ^{2}=36$
It can be written as
$4 x^{2}+9 y^{2}=36$
Or , $\frac{ x ^{2}}{9}+\frac{y^{2}}{4}=1$
Or, $\frac{x^{2}}{3^{2}}+\frac{y^{2}}{2^{2}}=1$ ......... $(1)$
Here, the denominator of $\frac{ x ^{2}}{3^{2}}$ is greater than the denominator of $\frac{y^{2}}{2^{2}}$
Therefore, the major axis is along the $x-$ axis, while the minor axis is along the $y-$ axis.
On comparing the given equation with $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1,$ we obtain $a=3$ and $b=2$
$\therefore c=\sqrt{a^{2}-b^{2}}=\sqrt{9-4}=\sqrt{5}$
Therefore,
The coordinates of the foci are $(\pm \sqrt{5}, \,0)$
The coordinates of the vertices are $(±3,\,0)$
Length of major axis $=2 a=6$
Length of minor axis $=2 b=4$
Eccentricity, $e=\frac{c}{a}=\frac{\sqrt{5}}{3}$
Length of latus rectum $=\frac{2 b^{2}}{a}=\frac{2 \times 4}{3}=\frac{8}{3}$
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