Weds April. 23, 2008
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MATH 32A: FIRST MIDTERM EXAMINATION
SOLUTIONS
Spring 2008 – Copyright Dr. Frederick Park
1
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1.
(20 points) Find a vector function that represents the curve C of intersection
of the surface
z
2
=
x
2
+
y
2
and the plane 2
z
= 1+
y
for
z
≥
0. Graph the curve
and indicate the direction in which the curve C is traced as the parameter t
increases from your parametrization of C.
Solution:
There are many different ways to do this.
The clearest way is to note that
if we set
z
=
c
= constant, then
x
2
+
y
2
=
z
2
=
c
2
defines a circle of radius
z
=
c
. Thus, cross sections of
x
2
+
y
2
=
z
2
wrt the xy plane are circles. If we
set
y
= 0, then
x
2
+
y
2
=
z
2
becomes
z
2
=
x
2
or

z

=

x

, since
z
≥
0, this
implies
z
=

x

. Similarly
z
=

y

when
x
= 0. Thus, the surface defined by
x
2
+
y
2
=
z
2
is a cone w/ vertex emanating from the origin.
Now, from conic sections, we know that a plane intersected in a non orthogo
nal manner is an ellipse. Thus
C
is an ellipse.
Now, if we set
x
=
z
cos
t
and
y
=
z
sin
t
. Then for
z
=
c
= constant, this
defines a circle of radius
z
2
=
c
2
oriented in the clockwise direction. If we plug
these into the given equation for the plane we obtain 2
z
= 1 +
y
= 1 +
z
sin
t
.
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 Spring '08
 GANGliu
 Math, Vector Space, Parametric equation, Conic section, Weds April.

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