Question 1074902
The center of that ellipse is the midpoint of the major axis,
half way between (0,0) and (13,0) on line y=0.
The coordinates of that midpoint are the averages of the coordinates of the vertices.
The midpoint is (6.5,0) , and that is the center of the ellipse.
The distance from the center to each vertex is called the semi-major axis and represented as {{{a}}} .
In this case {{{a=6.5}}} .
Half of the minor axis is called the semi-minor axis and is represented with {{{b}}} .
That is the distance from the center to each co-vertex.
In this case {{{b=12/2=6}}} .
The distance from the center to each focus is represented with {{{c}}} ,
and is related to {{{a}}} and {{{b}}} by {{{a^2=b^2+c^2}}} .
In this case {{{6.5^2=6^2+c^2}}} .
Solving for {{{c}}} :
{{{6.5^2-6^2=c^2}}}
A calculator can do that, but I don't need one
{{{(6.5+6))(6.5-6)=c^2}}}
{{{0.5*12.5=c^2}}}
{{{6.25=c^2}}}
{{{c=sqrt(6.25)}}}
{{{c=2.5}}} .
That is the distance from center (6.5,0) to each focus.
The foci are on the same line as the vertices, line y=0,
on either side of the center,
so the x-coordinate of one focus is {{{6.5-2.5=4}}} ,
and the x-coordinate of the other focus is {{{6.5+2.5=9}}} .
So, the foci are {{{highlight("( 4 , 0 ) , (9 , 0 )")}}} .