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

Answers (1)

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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