Identify an equation in standard form for an ellipse with its center at the origin, a vertex at (0, 12), and a focus at (0, 8).

Explanation:
From the given:
Foci:
F
1
(
0
,
8
)
,
F
2
(
0
,
−
8
)

Co-vertices at
(
8
,
0
)
and (-8, 0)#
by inspection, the center of the ellipse is at Origin
(
0
,
0
)

that means
Center
(
h
,
k
)
=
(
0
,
0
)
Also, by inspection,
c
=
8
the distance from the center to a focus.
Also,
b
=
8
the distance from the center to one end point of the minor axis also called semi-minor axis length.
Compute for semi-major axis length
a
:
a
2
=
b
2
+
c
2
a
=
√
b
2
+
c
2
a
=
√
8
2
+
8
2
=
√
64
+
64
a
=
√
2
⋅
64
a
=
8
√
2
Use now the standard Form of the equation of ellipse for Vertical Major Axis.
(
x
−
h
)
2
b
2
+
(
y
−
k
)
2
a
2
=
1
Let us put on the values of
h
=
0
,
k
=
0
,
a
=
8
√
2
,
b
=
8
(
x
−
0
)
2
8
2
+
(
y
−
0
)
2
(
8
√
2
)
2
=
1
x
2
64
+
y
2
128
=
1
Kindly check the graph
graph{(x^2/64+y^2/128-1)=0[-25,25,-15,15]}