SECTION
13.6
A Survey of Quadric Surfaces
(ET Section 12.6)
405
33.
Find the equation of the ellipsoid passing through the points marked in Figure 14(A).
z
y
x
(A)
6
4
−
4
2
−
2
−
6
z
y
x
(B)
4
−
4
2
−
2
FIGURE 14
SOLUTION
The equation of an ellipsoid is
³
x
a
´
2
+
³
y
b
´
2
+
³
z
c
´
2
=
1(
1
)
The
x
,
y
and
z
intercepts are
(
±
a
,
0
,
0
)
,
(
0
,
±
b
,
0
)
and
(
0
,
0
,
±
c
)
respectively. The
x
,
y
and
z
intercepts of the desired
ellipsoid are
(
±
2
,
0
,
0
)
,
(
0
,
±
4
,
0
)
and
(
0
,
0
,
±
6
)
respectively, hence
a
=
2,
b
=
4and
c
=
6. Substituting into (1) we
get
³
x
2
´
2
+
³
y
4
´
2
+
³
z
6
´
2
=
1
.
34.
Find the equation of the elliptic cylinder passing through the points marked in Figure 14(B).
The equation of the elliptic cylinder in the
xyz
-coordinate system is
³
x
a
´
2
+
³
y
b
´
2
=
1
)
The
x
and
y
intercepts are
(
±
a
,
0
)
and
(
0
,
±
b
)
respectively. The
x
and
y
intercepts of the desired cylinder are
(
±
2
,
0
)
and
(
0
,
±
4
)
respectively, hence
a
=
2and
b
=
4. Substituting into (1) we obtain the following equation:
³
x
2
´
2
+
³
y
4
´
2
=
1
.