Find the coordinates of the foci, the vertices, the lengths of major and minor axes and the eccentricity of the ellipse $9 x^{2}+4 y^{2}=36$.
Find the coordinates of the foci, the vertices, the lengths of major and minor axes and the eccentricity of the ellipse $9 x^{2}+4 y^{2}=36$.
The given equation of the ellipse can be written in standard form as
$\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$
since the denominator of $\frac{y^{2}}{9}$ is larger than the denominator of $\frac{x^{2}}{4},$ the major axis is along the $y-$ axis. Comparing the given equation with the standard equation
$\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$, we have $b=2$ and $a=3$
Also $c=\sqrt{a^{2}-b^{2}}$ $=\sqrt{9-4}=\sqrt{5}$
and $e=\frac{c}{a}=\frac{\sqrt{5}}{3}$
Hence the foci are $(0,\, \sqrt{5})$ and $(0,\,-\sqrt{5}),$ vertices are $(0,\,3)$ and $(0,\,-3),$ length of the major axis is $6$ units, the length of the minor axis is $4$ units and the eccentricity of the cllipse is $\frac{\sqrt{5}}{3}$.
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