આપેલ ઉપવલય માટે નાભિના યામ, શિરોબિંદુઓ તથા પ્રધાન અક્ષ તથા ગૌણ અક્ષની લંબાઈ, ઉત્કેન્દ્રતા અને નાભિલંબની લંબાઈ શોધોઃ

$\frac{x^{2}}{25}+\frac{y^{2}}{100}=1$

The given equation is $\frac{x^{2}}{25}+\frac{y^{2}}{100}=1$ or $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{10^{2}}=1$

Here, the denominator of $\frac{y^{2}}{100}$ is greater than the denominator of $\frac{x^{2}}{25}$

Therefore, the major axis is along the $y-$ axis, while the minor axis is along the $x-$ axis.

On comparing the given equation with $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1,$ we obtain $b=5$ and $a=10$

$\therefore c=\sqrt{a^{2}-b^{2}}=\sqrt{100-25}=\sqrt{75}=5 \sqrt{3}$

Therefore,

The coordinates of the foci are $(0, \,\pm 5 \sqrt{3})$

The coordinates of the vertices are $(0,\,±10)$

Length of major axis $=2 a=20$

Length of minor axis $=2 b=10$

Eccentricity, $e=\frac{c}{a}=\frac{5 \sqrt{3}}{10}=\frac{\sqrt{3}}{2}$

Length of latus rectum $=\frac{2 b^{2}}{a}=\frac{2 \times 25}{10}=5$