60.
Prove that
are parametric equations for a hyperbola. Assume
that
are nonzero constants.
61.
Consider the parametric curve
Assume that
a
and
b
are nonzero constants. Find the
Cartesian equation for this curve.
y
=
b
cos
2
t
+
a
sin
2
t
, 0
…
t
…
p
2
.
x
=
a
sin
2
t

b
cos
2
t
,
a
and
b
t
Z
p
, 3
p
0
…
t
…
4
p
,
y
=
b
cot
a
t
2
b
,
x
=
a
csc
a
t
2
b
,
a
and
b
3
p
2
t
Z
p
2
,
0
…
t
…
2
p
,
y
=
b
sec
t
,
x
=
a
tan
t
,
62.
Consider the parametric curve
Assume that
a
is not zero. Find the Cartesian equation for
this curve.
63.
Consider the parametric curve
Assume that
a
is a positive integer and
b
is a positive real
number. Determine the Cartesian equation.
64.
Consider the parametric curve
Assume that
b
is a positive integer and
a
is a positive real
number. Determine the Cartesian equation.
x
=
a
ln
t
,
y
=
ln(
bt
),
t
7
0.
x
=
e
at
,
y
=
be
t
,
t
7
0.
y
=
a
cos
t

a
sin
t
, 0
…
t
…
2
p
.
x
=
a
sin
t
+
a
cos
t
,
53.Find the rectangular equation that corresponds to the planecurve defined by the parametric equations andDescribe the plane curve.
1.
t
=
y
2
y
2
=
t
y
=
1
t
.
y
=
1
t
.
x
=
t
+
1
54.Find the rectangular equation that corresponds to the planecurve defined by the parametric equations andDescribe the plane curve.
1).
y
=
x
2

1
y
=
x
2

1
y
=
t

1.
t
=
x
2
x
2
=
t
x
=
1
t
.
y
=
t

1.
x
=
1
t
In Exercises 53 and 54, explain the mistake that is made.
■
C ATC H T H E M I S TA K E
■
C O N C E P T U A L
In Exercises 55 and 56, determine whether each statement is true or false.
■
C H A L L E N G E
■
T E C H N O L O GY
“And the Rockets Red Glare . . .”
Scientists at Vandenberg Air Force Base are interested in tracing the path of
some newly designed rockets. They will launch two rockets at 100 feet per second.
One will depart at 45 , the other at 60 . From vector analysis and gravity, you
determine the following coordinates (
x
,
y
) as a function of time
t
where
y
stands
for height in feet above the ground and
x
stands for lateral distance traveled.
45
60
x
100 cos(45 )
t
x
100 cos(60 )
t
y
16
t
2
100 sin(45 )
t
y
16
t
2
100 sin(60 )
t
1.
For each angle (45 , 60 ), fill in the chart (round to one decimal place). You can
do this by hand (very slowly) or use the table capabilities of a calculator or similar
device.
CHAPTER 11
INQUIRYBASED LEARNING PROJECT
t
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
X
45
Y
45
X
60
Y
60
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 Summer '17
 juan alberto
 Parametric Equations, Conic section