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
²
·
±
x
,
y
,
z
²
=
4
or
±
2
3
,

1
3
,
2
3
²
·
±
x
,
y
,
z
²
=
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
²
=
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
²
=
O P
=
n
in
the equation of the plane gives
n
·
±
x
,
y
,
z
²
=
n
·
O P
=
n
·
n
=
³
n
³
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
=
³
1
√
3
,
1
√
3
,
1
√
3
´
.
We substitute
n
=
O P
=
µ
1
√
3
,
1
√
3
,
1
√
3
¶
and
R
=
³
O P
³
=
1 in the equation
n
·
±
x
,
y
,
z
²
=
R
2
of the
plane tangent to the unit sphere at
P
. This gives
±
1
√
3
,
1
√
3
,
1
√
3
²
·
±
x
,
y
,
z
²
=
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.
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.