# 4. 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.       $\frac{x^2}{25} + \frac{y^2}{100} = 1$

Given

The equation of the ellipse

$\frac{x^2}{25} + \frac{y^2}{100} = 1$

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

$a=10$ and $b=5$.

So,

$c=\sqrt{a^2-b^2}=\sqrt{10^2-5^2}$

$c=\sqrt{75}=5\sqrt{3}$

Hence,

Coordinates of the foci:

$(0,c)\:and\:(0,-c)=(0,5\sqrt{3})\:and\:(0,-5\sqrt{3})$

The vertices:

$(0,a)\:and\:(0,-a)=(0,10)\:and\:(0,-10)$

The length of the major axis:

$2a=2(10)=20$

The length of minor axis:

$2b=2(5)=10$

The eccentricity :

$e=\frac{c}{a}=\frac{5\sqrt{3}}{10}=\frac{\sqrt{3}}{2}$

The length of the latus rectum:

$\frac{2b^2}{a}=\frac{2(5)^2}{10}=\frac{50}{10}=5$

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