Find the value

Given:

$\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$ ..........(i)

Since, $9<16$

So, above equation is of the form,

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

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

$a^{2}=16$ and $b^{2}=9$

$\Rightarrow a=\sqrt{16}$ and $b=\sqrt{9}$

$\Rightarrow a=4$ and $b=3$

(i) To find: Length of major axes

Clearly, $a

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

$=2 \times 4$

$=8$ units

(ii) To find: Coordinates of the Vertices

Clearly, $a>b$

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

$=(0,4)$ and $(0,-4)$

(iii) To find: Coordinates of the foci

We know that,

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

So, firstly we find the value of $c$

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

$=16-9$

$c^{2}=7$

$c=\sqrt{7} \ldots(l)$

$\therefore$ Coordinates of foci $=(0, \pm \sqrt{7})$

(iv) To find: Eccentricity

We know that,

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

$\Rightarrow \mathrm{e}=\frac{\sqrt{7}}{4}[$ from $(\mathrm{I})]$

(v) To find: Length of the Latus Rectum

We know that,

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

$=\frac{2 \times(3)^{2}}{4}$

$=\frac{9}{2}$