Find the equation of an ellipse with vertices (0,±8) and foci (0,±4).
The equation of an ellipse is (x−h)2a2+(y−k)2b2=1 for a horizontally oriented ellipse and (x−h)2b2+(y−k)2a2=1 for a vertically oriented ellipse.
(h,k) is the center and the distance c from the center to the foci is given by a2−b2=c2. a is the distance from the center to the vertices and b is the distance from the center to the co-vertices.
The center of the ellipse is half way between the vertices. Thus, the center (h,k) of the ellipse is (0,0) and the ellipse is vertically oriented.
a is the distance between the center and the vertices, so a=8.
c is the distance between the center and the foci, so c=4
a2−b2=c2⇒b2=a2−c2
b2=82−42=64−16=48
The equation is:
(x−0)248+(y−0)264=1 or x248+y264=1
