Gauthier, Joseph – Homework 2 – Due: Sep 6 2007, 3:00 am – Inst: JEGilbert
2
2.
Moves once counterclockwise along the
ellipse
x
2
4
+
y
2
16
= 1
,
starting and ending at (0
,
4).
3.
Moves once clockwise along the ellipse
x
2
4
+
y
2
16
= 1
,
starting and ending at (0
,
4).
correct
4.
Moves once counterclockwise along the
circle
(2
x
)
2
+ (4
y
)
2
= 1
,
starting and ending at (0
,
4).
5.
Moves along the line
x
2
+
y
4
= 1
,
starting at (0
,
4) and ending at (2
,
0).
6.
Moves once clockwise along the circle
(2
x
)
2
+ (4
y
)
2
= 1
,
starting and ending at (0
,
4).
Explanation:
Since
cos
2
t
+ sin
2
t
= 1
for all
t
, the particle travels along the curve
given in Cartesian form by
x
2
4
+
y
2
16
= 1 ;
this is an ellipse centered at the origin.
At
t
= 0, the particle is at (2 sin 0
,
4 cos 0),
i.e.
,
at the point (0
,
4) on the ellipse.
Now as
t
increases from
t
= 0 to
t
=
π/
2,
x
(
t
) increases
from
x
= 0 to
x
= 2, while
y
(
t
) decreases from
y
= 4 to
y
= 0 ; in particular, the particle
moves from a point on the positive
y
-axis to
a point on the positive
x
-axis, so it is moving
clockwise
.
In the same way, we see that as
t
increases
from
π/
2 to
π
, the particle moves to a point
on the negative
y
-axis, then to a point on the
negative
x
-axis as
t
increases from
π
to 3
π/
2,
until Fnally it returns to its starting point on
the positive
y
-axis as
t
increases from 3
π/
2 to
2
π
.
Consequently, the particle moves clockwise