दीर्घवृत्त $9 x^{2}+4 y^{2}=36$ के नाभियों और शीर्षों के निर्देशांक, दीर्घ और लघु अक्ष की लंबाइयाँ, और उत्केंद्रता ज्ञात कीजिए।

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 $\frac{\sqrt{5}}{3}$.