दीर्घवृत्त में नाभियों और शीर्षों के निर्देशांक, दीर्घ और लघु अक्ष की लंबाइयाँ, उत्केंद्रता तथा नाभिलंब जीवा की लंबाई ज्ञात कीजिए

$4 x ^{2}+9 y ^{2}=36$

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 $=\frac{2 b^{2}}{a}=\frac{2 \times 4}{3}=\frac{8}{3}$