Answer:
The given equation is
![25x^{2} +4y^{2}=100](https://tex.z-dn.net/?f=25x%5E%7B2%7D%20%2B4y%5E%7B2%7D%3D100)
Which represents an elipse.
To find its elements, we need to divide the equation by 100
![\frac{25x^{2} +4y^{2} }{100} =\frac{100}{100} \\\frac{x^{2} }{4} +\frac{y^{2} }{25} =1](https://tex.z-dn.net/?f=%5Cfrac%7B25x%5E%7B2%7D%20%2B4y%5E%7B2%7D%20%7D%7B100%7D%20%3D%5Cfrac%7B100%7D%7B100%7D%20%5C%5C%5Cfrac%7Bx%5E%7B2%7D%20%7D%7B4%7D%20%2B%5Cfrac%7By%5E%7B2%7D%20%7D%7B25%7D%20%3D1)
Where
and
. Remember that the greatest denominator is
, and the least is
. So, we extract the square root on each equation.
and
.
In a elipse, we have a major axis and a minor axis. In this case, the major axis is vertical and the minor axis is horizontal, that means this is a vertical elipse.
The length of the major axis is
.
The length of the minor axis is
.
The vertices are
and
.
Now, the main parameters of an elipse are related by
, which we are gonna use to find
, the parameter of the focus.
![c=\sqrt{a^{2}-b^{2} }=\sqrt{25-4}=\sqrt{21}](https://tex.z-dn.net/?f=c%3D%5Csqrt%7Ba%5E%7B2%7D-b%5E%7B2%7D%20%7D%3D%5Csqrt%7B25-4%7D%3D%5Csqrt%7B21%7D)
So, the coordinates of each focus are
and ![(0,-\sqrt{21})](https://tex.z-dn.net/?f=%280%2C-%5Csqrt%7B21%7D%29)
The eccentricity of a elipse is defined
![e=\frac{c}{a}=\frac{\sqrt{21} }{5} \approx 0.92](https://tex.z-dn.net/?f=e%3D%5Cfrac%7Bc%7D%7Ba%7D%3D%5Cfrac%7B%5Csqrt%7B21%7D%20%7D%7B5%7D%20%20%5Capprox%200.92)
The latus rectum is defined
![L=\frac{2b^{2} }{a}=\frac{2(4)}{5} =\frac{8}{5} \approx 1.6](https://tex.z-dn.net/?f=L%3D%5Cfrac%7B2b%5E%7B2%7D%20%7D%7Ba%7D%3D%5Cfrac%7B2%284%29%7D%7B5%7D%20%3D%5Cfrac%7B8%7D%7B5%7D%20%5Capprox%201.6)
Finally, the graph of the elipse is attached.