# 8. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.       $16x^2 + y^2 = 16$

Given

The equation of the ellipse

$16x^2 + y^2 = 16$

$\frac{16x^2}{16} + \frac{y^2}{16} = 1$

$\frac{x^2}{1^2} + \frac{y^2}{4^2} = 1$

As we can see from the equation, the major axis is along Y-axis and the minor axis is along X-axis.

On comparing the given equation with the standard equation of such  ellipse, which is

$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$

We get

$a=4$ and $b=1$.

So,

$c=\sqrt{a^2-b^2}=\sqrt{4^2-1^2}$

$c=\sqrt{15}$

Hence,

Coordinates of the foci:

$(0,c)\:and\:(0,-c)=(0,\sqrt{15})\:and\:(0,-\sqrt{15})$

The vertices:

$(0,a)\:and\:(0,-a)=(0,4)\:and\:(0,-4)$

The length of the major axis:

$2a=2(4)=8$

The length of minor axis:

$2b=2(1)=2$

The eccentricity :

$e=\frac{c}{a}=\frac{\sqrt{15}}{4}$

The length of the latus rectum:

$\frac{2b^2}{a}=\frac{2(1)^2}{4}=\frac{2}{4}=\frac{1}{2}$

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