S E C T I O N
13.6
A Survey of Quadric Surfaces
(ET Section 12.6)
397
or
1
6
4
,

2
,
4
·
x
,
y
,
z
=
4
or
2
3
,

1
3
,
2
3
·
x
,
y
,
z
=
4
,
in normal form.
71.
Let
n
=
→
O P
, where
P
=
(
x
0
,
y
0
,
z
0
)
is a point on the sphere
x
2
+
y
2
+
z
2
=
r
2
, and let
P
be the plane
with equation
n
·
x
,
y
,
z
=
r
2
. Show that the point on
P
nearest the origin is
P
itself and conclude that
P
is tangent to
the sphere at
P
(Figure 13).
FIGURE 13
The terminal point of
n
lies on the sphere of radius
r
.
SOLUTION
First notice that the terminal point
P
of
n
lies on the plane
P
, since substituting
x
,
y
,
z
=
→
O P
=
n
in
the equation of the plane gives
n
·
x
,
y
,
z
=
n
·
→
O P
=
n
·
n
=
n
2
=
r
2
Since the vector
→
O P
=
n
is orthogonal to the plane, that is, the radius
O P
is perpendicular to the plane, we conclude
that
P
is the point on
P
closest to the origin and that
P
is tangent to the sphere of radius
r
centered at the origin. Of
course,
P
is the tangency point.
72.
Use Exercise 71 to find the equation of the plane tangent to the unit sphere at
P
=
1
√
3
,
1
√
3
,
1
√
3
.
SOLUTION
We substitute
n
=
→
O P
=
1
√
3
,
1
√
3
,
1
√
3
and
R
=
→
O P
=
1 in the equation
n
·
x
,
y
,
z
=
R
2
of the
plane tangent to the unit sphere at
P
. This gives
1
√
3
,
1
√
3
,
1
√
3
·
x
,
y
,
z
=
1
1
√
3
x
+
1
√
3
y
+
1
√
3
z
=
1
x
+
y
+
z
=
√
3
13.6 A Survey of Quadric Surfaces
(ET Section 12.6)
Preliminary Questions
1.
True or false: All traces of an ellipsoid are ellipses.
SOLUTION
This statement is true, mostly. All traces of an ellipsoid
(
x
a
)
2
+
(
y
b
)
2
+
(
z
c
)
2
=
1 are ellipses, except for
the traces obtained by intersecting the ellipsoid with the planes
x
= ±
a
,
y
= ±
b
and
z
= ±
c
. These traces reduce to the
single points
(
±
a
,
0
,
0
)
,
(
0
,
±
b
,
0
)
and
(
0
,
0
,
±
c
)
respectively.
2.
True or false: All traces of a hyperboloid are hyperbolas.
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C H A P T E R
13
VECTOR GEOMETRY
(ET CHAPTER 12)
SOLUTION
The statement is false. For a hyperbola in the standard orientation, the horizontal traces are ellipses (or
perhaps empty for a hyperbola of two sheets), and the vertical traces are hyperbolas.
3.
Which quadric surfaces have both hyperbolas and parabolas as traces?
SOLUTION
The hyperbolic paraboloid
z
=
(
x
a
)
2

(
y
b
)
2
has vertical trace curves which are parabolas. If we set
x
=
x
0
or
y
=
y
0
we get
z
=
x
0
a
2

y
b
2
⇒
z
=

y
b
2
+
C
z
=
x
a
2

y
0
b
2
⇒
z
=
x
a
2
+
C
The hyperbolic paraboloid has vertical traces which are hyperbolas, since for
z
=
z
0
, (
z
0
>
0), we get
z
0
=
x
a
2

y
b
2
4.
Is there any quadric surface whose traces are all parabolas?
SOLUTION
There is no quadric surface whose traces are all parabolas.
5.
A surface is called
bounded
if there exists
M
>
0 such that every point on the surfaces lies at a distance of at most
M
from the origin. Which of the quadric surfaces are bounded?
SOLUTION
The only quadric surface that is bounded is the ellipsoid
x
a
2
+
y
b
2
+
z
c
2
=
1
.
All other quadric surfaces are not bounded, since at least one of the coordinates can increase or decrease without bound.
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 Spring '08
 Hubscher
 Conic section, Hyperboloid, Quadric

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