romasko (qrr58) – Assignment 1 – luecke – (55035)
2
which after expansion and simplification can
be rewritten as
9
x
2
−
18
x
+ 4
y
2
−
8
y
= 23
.
keywords:
graph,
conic
section,
ellipse,
shifted ellipse
003
10.0 points
Which one of the following conic sections
has graph
2
4
−
2
−
4
−
6
2
4
−
2
−
4
−
6
1.
x
2
−
2
x
+ 4
y
2
−
8
y
+ 1 = 0
2.
x
2
+ 2
x
−
4
y
2
+ 8
y
= 7
3.
x
2
−
2
x
−
4
y
2
+ 8
y
= 7
4.
4
y
2
−
8
x
−
x
2
−
2
x
= 1
5.
x
2
+ 2
x
+ 4
y
2
+ 8
y
+ 1 = 0
6.
x
2
+ 2
x
−
4
y
2
−
8
y
= 7
7.
4
y
2
+ 8
x
−
x
2
−
2
y
= 1
8.
4
y
2
+ 8
y
−
x
2
−
2
x
= 1
correct
Explanation:
The graph is a shifted hyperbola having
asymptotes which intersect at (
−
1
,
−
1) as
well as vertices at (
−
1
,
0) and (
−
1
,
−
2). Thus
one equation for this hyperbola is
(
y
+ 1)
2
1
−
(
x
+ 1)
2
4
= 1
,
which after expansion and simplification can
be rewritten as
4
y
2
+ 8
y
−
x
2
−
2
x
= 1
.
keywords:
graph,
conic
section,
ellipse,
shifted ellipse
004
10.0 points
Which one of the points
P
(0
,
−
2
,
0)
,
Q
(
−
3
,
9
,
1)
,
R
(2
,
−
5
,
3)
in 3space is closest to the
xz
plane?
1.
P
(0
,
−
2
,
0)
correct
2.
R
(2
,
−
5
,
3)
3.
Q
(
−
3
,
9
,
1)
Explanation:
The distance of a point (
a, b, c
) in 3space
from the
xz
plane is given by

b

.
Conse
quently, of the three points
P
(0
,
−
2
,
0)
,
Q
(
−
3
,
9
,
1)
,
R
(2
,
−
5
,
3)
the one closest to the
xz
plane is
P
(0
,
−
2
,
0)
.
keywords: plane, distance in 3space,
005
10.0 points
Which of the following statements are true:
A.
in
2
space the intersection of the graphs of
braceleftbigg
x
= 3
,
y
= 5
,
is a line
,