vertices (0,±4) and co vertices (±1,0)
Answers
Answer:
Find the equation of an ellipse with vertices
(
0
,
±
8
)
and foci
(
0
,
±
4
)
.
The equation of an ellipse is
(
x
−
h
)
2
a
2
+
(
y
−
k
)
2
b
2
=
1
for a horizontally oriented ellipse and
(
x
−
h
)
2
b
2
+
(
y
−
k
)
2
a
2
=
1
for a vertically oriented ellipse.
(
h
,
k
)
is the center and the distance
c
from the center to the foci is given by
a
2
−
b
2
=
c
2
.
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
a
2
−
b
2
=
c
2
⇒
b
2
=
a
2
−
c
2
b
2
=
8
2
−
4
2
=
64
−
16
=
48
The equation is:
(
x
−
0
)
2
48
+
(
y
−
0
)
2