Question 1037746
So, that is E, but what is the question?
Write the equation?
Locate vertices and co-vertices?
graph?
 

Both foci are on the vertical line {{{x=4}}} ,
so this is an ellipse with foci and vertices on the vertical major axis {{{x=4}}} .
The center is the midpoint of the segment connecting the foci.
That is point (4,3), with coordinates
{{{x=(4+4)/2=4}}} and {{{y=(-2+8)/2=6/2=3}}}
the averages of the foci coordinates.
The focal distance, {{{c}}} , is the distance from one focus to the center:
{{{c=8-3=5}}} .
Since the major axis has length {{{20}}} ,
the semi-major axis is {{{a=20/2=10}}} .
Since the major axis is vertical, the vertices of the ellipse are {{{c=10}}} units above and below center (4,3), at (4,-7) and (4,13).
The semi-minor axis, {{{b}}} , can be calculated using the relation
{{{a^2=b^2+c^2}}} .
{{{10^2=3^2+c^2}}}
100-9=b^2}}}
{{{b^2=91}}}
{{{b=sqrt(21)}}}
The equation of an ellipse with major axis parallel to the y-axis,
center at point (h,k), is
{{{(x-h)^2/b^2+(y-k)^2/a^2=1}}}
In this case, with {{{system(h=4,k=3,a=10,b=sqrt(91))}}} ,
the equation is
{{{(x-a)^2/91+(y-3)^2/100=1}}} .
The co-vertices are on the horizontal minor axis,
a distance {{{b=sqrt(91)}}} to the left and rihjt of center (4,3),
at {{{"("}}}{{{4-sqrt(91)}}}{{{", 3 )"}}} and {{{"("}}}{{{4+-sqrt(91)}}}{{{", 3 )"}}} .