Find the value

Given:

$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$

…(i)

Since, 25 > 9

So, above equation is of the form, 

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$     ...........(ii)

Comparing eq. (i) and (ii), we get

$a^{2}=25$ and $b^{2}=9$

$\Rightarrow a=\sqrt{2} 5$ and $b=\sqrt{9}$

$\Rightarrow a=5$ and $b=3$

(i) To find: Length of major axes

Clearly, $a>b$, therefore the major axes of the ellipse is along $x$ axes.

$\therefore$ Length of major axes $=2 \mathrm{a}$

$=2 \times 5$

$=10$ units

(ii) To find: Coordinates of the Vertices

Clearly, $a>b$

$\therefore$ Coordinate of vertices $=(a, 0)$ and $(-a, 0)$

$=(5,0)$ and $(-5,0)$

(iii) To find: Coordinates of the foci

We know that,

Coordinates of foci $=(\pm c, 0)$ where $c^{2}=a^{2}-b^{2}$

So, firstly we find the value of c

$c^{2}=a^{2}-b^{2}$

$=25-9$

$c^{2}=16$

$c=\sqrt{16}$

$c=4 \ldots(I)$

$\therefore$ Coordinates of foci $=(\pm 4,0)$

(iv) To find: Eccentricity

We know that,

Eccentricity $=\frac{\mathrm{c}}{\mathrm{a}}$

$\Rightarrow \mathrm{e}=\frac{4}{5}[$ from $(\mathrm{I})]$

(v) To find: Length of the Latus Rectum

We know that,

Length of Latus Rectum $=\frac{2 b^{2}}{a}$

$=\frac{2 \times(3)^{2}}{5}$

$=\frac{18}{5}$