S E C T I O N
13.6
A Survey of Quadric Surfaces
(ET Section 12.6)
409
Combining (3) and (4) gives
u
1
=
0
,
y
(
t
0
),
−
c
2
n
b
2
r
y
(
t
0
)
=
y
(
t
0
)
0
,
1
,
−
c
2
n
b
2
r
u
2
=
x
(
t
1
),
0
,
−
c
2
m
a
2
r
x
(
t
1
)
=
x
(
t
1
)
1
,
0
,
−
c
2
m
a
2
r
u
3
=
x
(
t
2
),
−
b
2
m
a
2
n
x
(
t
2
),
0
=
x
(
t
2
)
1
,
−
b
2
m
a
2
n
,
0
(c)
Now, to show that the tangent lines are contained in
P
, we must show that their direction vectors are in
P
, that is,
they are orthogonal to the vector
n
=
m
a
2
,
n
b
2
,
r
c
2
normal to the plane. We show this by proving that the following dot
products are zero. That is,
u
1
·
n
=
y
(
t
0
)
0
,
1
,
−
c
2
n
b
2
r
·
m
a
2
,
n
b
2
,
r
c
2
=
y
(
t
0
)
n
b
2
−
n
b
2
=
0
u
2
·
n
=
x
(
t
1
)
1
,
0
,
−
c
2
m
a
2
r
·
m
a
2
,
n
b
2
,
r
c
2
=
x
(
t
1
)
m
a
2
−
m
a
2
=
0
u
3
·
n
=
x
(
t
2
)
1
,
−
b
2
m
a
2
n
,
0
·
m
a
2
,
n
b
2
,
r
c
2
=
x
(
t
2
)
m
a
2
−
m
a
2
=
0
Since the tangent lines intersect
P
(at the point
P
) and their direction vectors are in
P
, it follows that the tangent lines
are contained in the plane
P
.
42.
Let
S
be the hyperboloid
x
2
+
y
2
=
z
2
+
1 and let
P
=
(
α
,
β
,
0
)
be a point on
S
in the
(
x
,
y
)
plane. Show that there
are precisely two lines through
P
entirely contained in
S
(Figure 17).
Hint:
Consider the line
r
(
t
)
=
α
+
at
,
β
+
bt
,
t
through
P
. Show that
r
(
t
)
is contained in
S
if
(
a
,
b
)
is one of the two points on the unit circle obtained by rotating
(
α
,
β
)
through
±
π
2
. This proves that a hyperboloid of one sheet is a
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 Winter '08
 GANGliu
 Parametric equation, Conic section, 0 m, elliptic cone, E C T O R G E O M E T

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