Question 951643
For an ellipse with major axis horizontal:
{{{(x-h)^2/a^2+(y-k)^2/b^2=1}}}
Where (h,k) is the center,
a is distance from center to vertex,
b is distance from center to co-vertex,
The co-vertex is the intersection of the ellipse and the minor axis.
b can also be calculated from the equation {{{b^2=a^2-c^2}}} where a is distance from center to vertex, c is distance from center to focus.
In this case:
a=13 (from (3,5) to (-10, 5) the distance from 3 to -10=13
c=5  (from (3,5) to (8,5) the distance from 3 to 8=5 
b=sqrt(a^2-c^2)}}}={{{sqrt(13^2-5^2)=sqrt(144)}}}=12
The equation becomes:
{{{(x-3)^2/169+(y-5)^2/144=1}}}