Q

# 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. x ^2 / 16 + y ^2 / 9 = 1

3.  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.

$\frac{x^2}{16} + \frac{y^2}{9} = 1$

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Given

The equation of the ellipse

$\frac{x^2}{16} + \frac{y^2}{9} = 1$

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

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

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

We get

$a=4$ and $b=3$.

So,

$c=\sqrt{a^2-b^2}=\sqrt{4^2-3^2}$

$c=\sqrt{7}$

Hence,

Coordinates of the foci:

$(c,0)\:and\:(-c,0)=(\sqrt{7},0)\:and\:(-\sqrt{7},0)$

The vertices:

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

The length of the major axis:

$2a=2(4)=8$

The length of minor axis:

$2b=2(3)=6$

The eccentricity :

$e=\frac{c}{a}=\frac{\sqrt{7}}{4}$

The length of the latus rectum:

$\frac{2b^2}{a}=\frac{2(3)^2}{4}=\frac{18}{4}=\frac{9}{2}$

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