Answer:
The equation of the ellipse is
[tex]\frac{(x-2)^{2} }{100} +\frac{(y-1)^{2} }{36} =1[/tex]
This is the explicit form of an ellipse, where [tex]h=2[/tex] and [tex]k=1[/tex], which means the center of the ellipse is at [tex]C(2,1)[/tex].
Remember that the greatest denominator in an ellipse is the parameter [tex]a^{2}[/tex] and the least is [tex]b^{2}[/tex], so
[tex]a^{2}=100 \implies a=10\\ b^{2}=36 \implies b=6[/tex]
The length of the major axis is: [tex]2a=2(10)=20[/tex]
The length of the minor axis is: [tex]2b=2(6)=12[/tex]
Vertices are [tex](-8,1)[/tex] and [tex](12,1)[/tex], because the center is not the origin of the coordinate system so, the vertices are displaced. The least axis vertices are: [tex](2,7)[/tex] and [tex](2,-5)[/tex].
The foci are on the major axis, so their vertical coordinate is 1, their horizontal coordinate depends on parameter [tex]c[/tex], which is related as the following
[tex]a^{2}=b^{2} +c^{2} \\100-36=c^{2}\\ c=\sqrt{64}\\ c=8[/tex]
Therefore, the foci coordinates are [tex]F(10,1)[/tex] and [tex]F'(-6,1)[/tex].
The ladus rectus is
[tex]LR=\frac{2b^{2} }{a}=\frac{2(36)}{10}=7.2[/tex]
The eccentricity is
[tex]e=\frac{c}{a}=\frac{8}{10}=0.8[/tex]
Additionally, the graph is attached, there you can observe some elements of the ellipse.
(Remember, all elements of an ellipse depend on the three parameters a, b and c, that's what you need to find first)
