MATH 155/GRACEY
CH. 10 PRACTICE
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
At the given point, find the line that is normal to the curve at the given point.
1) y4 + x3 = y2 + 10x, normal at (0, 1)
2) 3x2 y - π cos y = 4π, normal at (1, π)
1)
2)
Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify the
particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph
traced by the particle and the direction of motion.
3) x = 3 cos t, y = 3 sin t, π ≤ t ≤ 2π
3)
5
y
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
4) x = 36t2 , y = 6t, -∞ ≤ t ≤ ∞
4)
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
1
5) x = 2t + 4, y = 6t + 6, -∞ ≤ t ≤ ∞
5)
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
6) x = 4 sin t, y = 2 cos t, 0 ≤ t ≤ 2π
5
6)
y
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
Find a parametrization for the curve.
7) The ray (half line) with initial point (-1, -9) that passes through the point (4, -16)
8) The upper half of the parabola x + 7 = y2
7)
8)
2
Graph the pair of parametric equations with the aid of a graphing calculator.
9) x = 5(t - sin t), y= 5(1 - cos t), 0 ≤ t ≤ 4π
9)
y
12
8
4
10
20
30
40
50
x
60
10) x = 6 cos t + 2 cos 3t, y = 6 sin t - 2 sin3t, 0 ≤ t ≤ 2π
10
10)
y
8
6
4
2
-10 -8
-6
-4
-2
2
4
6
8
10 x
-2
-4
-6
-8
-10
Plot the point whose polar coordinates are given.
11) (2, -5π/4)
11)
5
-5
5
-5
3
12) (2, π/2)
12)
5
-5
5
-5
13) (-4, 0)
13)
5
-5
5
-5
14) (-2, 2π/3)
14)
5
-5
5
-5
Determine if the given polar coordinates represent the same point.
15) (3, π/4), (-3, π/4)
16) (8, π/2), (-8, 3π/2)
15)
16)
4
17) (3, π/4), (3, 5π/4)
17)
18) (r, θ), (-r, θ + π)
18)
19) (6, π/6), (6, 7π/6)
19)
Find the Cartesian coordinates of the given point.
1
20) -4, π
2
20)
21) ( 3, π/6)
21)
22) (-4, -π/3)
22)
23) (-4, π)
23)
24) 14 2 ,
π
4
24)
25) -5, 5π/3
26) 18, -
25)
π
4
26)
Replace the polar equation with an equivalent Cartesian equation.
27) r cos θ = 12
28) 3r cos θ + 9r sin θ = 1
29) r =
27)
28)
1
4 cos θ - 5 sin θ
29)
30) r = 8 cot θ csc θ
30)
31) r = -5 csc θ
31)
32) r2 sin 2θ = 22
32)
33) r = 36 sin θ
33)
34) r2 = 20r cos θ
34)
35) r2 = 22r cos θ - 6r sin θ - 9
35)
5
Graph the polar equation.
36) r = 3(2 + 2 sin θ)
36)
10
5
-10
-5
5
r
10
-5
-10
37) r = 2 - cos θ
37)
10
5
-10
-5
5
r
10
-5
-10
38) r = 4 sin 4θ
38)
10
5
-10
-5
5
10
r
-5
-10
6
39) r = 4 cos 5θ
39)
10
5
-10
-5
5
10
r
-5
-10
40) r = 3(1 -sin θ)
40)
10
5
-10
-5
5
10
r
-5
-10
41) r = 2θ
41)
20
10
-20
-10
10
20 r
-10
-20
7
42) r = - 1 - sin θ
42)
4
3
2
1
-4
-3
-2
-1
1
2
3
4 r
-1
-2
-3
-4
43) r = 2 + cos θ
43)
4
3
2
1
-4
-3
-2
-1
1
2
3
4 r
-1
-2
-3
-4
Find the area of the specified region.
44) Inside the cardioid r = α(1 + sin θ), α > 0
44)
45) Inside one leaf of the four-leaved rose r = 7 sin 2θ
45)
46) Inside the limacon r = 8 + 2 sin θ
46)
47) Inside the smaller loop of the limacon r = 5 + 10 sin θ
47)
48) Inside the circle r =
48)
3 sin θ and outside the cardioid r = 1 + cos θ
49) Shared by the circles r = 2 and r = 4 cos θ
49)
50) Shared by the circle r = 5 and the cardioid r = 5(1 + sin θ)
50)
51) Inside the circle r = 6 and to the right of the line r = 3 sec θ
51)
52) Inside the circle r = 8 cos θ + 3 sin θ
52)
53) Inside the outer loop and outside the inner loop of the limacon r = 4 sin θ - 2
53)
8
Graph the parabola.
54) y = 2x2
54)
y
10
5
-10
-5
5
10
x
-5
-10
55) x = 3y2
55)
y
5
-10
-5
5
10
x
-5
Find the focus and directrix of the parabola.
1 2
56) x =y
12
56)
57) y2 - 28x = 0
57)
9
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Choose the equation that matches the graph.
58)
58)
y
20
10
-4
-2
2
4
x
-10
-20
A) 4x2 = -y
B) 4x2 = y
C) 4y2 = x
D) x2 = 4y
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Graph.
59) 5x2 + 25y2 = 225
59)
y
10
5
-10
-5
5
10
x
-5
-10
60) x2 + 16y2 = 16
60)
y
5
-5
5
x
-5
10
Find the vertices and foci of the ellipse.
61) 16x2 + 121y2 = 1936
61)
Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions.
62) An ellipse with length of major axis 14 and y-intercepts (0, ±2)
62)
63) An ellipse with vertices (0, ±9) and foci at (0, ±2 3).
63)
Graph.
64)
y2 x2
=1
16 49
64)
y
10
5
-10
-5
5
10
x
-5
-10
65) 16y2 - 4x2 = 64
65)
y
10
5
-10
-5
5
10
x
-5
-10
Solve the problem.
66) Find the foci and asymptotes of the following hyperbola:
x2 y2
=1
64 36
67) Find the vertices and asymptotes of the following hyperbola:
y2
x2
=1
100 64
11
66)
67)
Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions.
68) Vertices at (2, 0) and (-2, 0); foci at (3, 0) and (-3, 0)
68)
69) Vertices at (0, 6) and (0, -6); asymptotes y =
3
3
x and y = - x
5
5
69)
Find an equation of the following curve.
70) An ellipse centered at the origin having vertex at (0, -3) and eccentricity equal to
1
3
Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin.
20
71) r =
5 + 5 cos θ
70)
71)
6
4
2
-6
-4
-2
2
4
6
-2
-4
-6
72) r =
20
5 + 5 sin θ
72)
6
4
2
-6
-4
-2
2
4
6
-2
-4
-6
12
73) r =
8
2 - cos θ
73)
10
5
-10
-5
5
10
-5
-10
13
Answer Key
Testname: M155_E4_PRACTICE
1) y = 2) y =
1
x+1
5
1
1
x+π
2π
2π
3) x2 + y2 = 9
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
Counterclockwise from (-3, 0) to (3, 0)
4) x = y2
5
y
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
Entire parabola, bottom to top (from fourth quadrant to origin to first quadrant)
14
Answer Key
Testname: M155_E4_PRACTICE
5) y = 3x - 6
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
3
4
5 x
-1
-2
-3
-4
-5
Entire line, from left to right
x2 y2
6)
+
=1
16
4
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
-1
-2
-3
-4
-5
Counterclockwise from (0, 2) to (0, 2), one rotation
7) Answers will vary. Possible answer: x = -1 + 5t, y = -9 - 7t, t ≥ 0
8) Answers will vary. Possible answer: x = t2 + 7, y = t, t ≥ 0
9)
y
12
8
4
10
20
30
40
50
60
x
15
Answer Key
Testname: M155_E4_PRACTICE
10)
10
y
8
6
4
2
-10 -8
-6
-4
-2
2
4
6
8
10 x
-2
-4
-6
-8
-10
11)
5
-5
5
-5
12)
5
-5
5
-5
16
Answer Key
Testname: M155_E4_PRACTICE
13)
5
-5
5
-5
14)
5
-5
5
-5
15)
16)
17)
18)
19)
20)
No
Yes
No
Yes
No
(0, -4)
3
3
21) ,
2 2
22) (-2, 2 3)
23) (4, 0)
24) (14, 14)
-5 5 3
25)
,
2
2
(9 2, -9 2)
x = 12
3x + 9y = 1
4x - 5y = 1
30) y2 = 8x
31) y = -5
26)
27)
28)
29)
17
Answer Key
Testname: M155_E4_PRACTICE
32) y =
11
x
33) x2 + (y - 18)2 = 324
34) (x - 10)2 + y2 = 100
35) (x - 11)2 + (y + 3)2 = 121
36)
10
5
-10
-5
5
10
r
5
10
r
-5
-10
37)
10
5
-10
-5
-5
-10
38)
10
5
-10
-5
5
10
r
-5
-10
18
Answer Key
Testname: M155_E4_PRACTICE
39)
10
5
-10
-5
5
10
r
5
10
r
-5
-10
40)
10
5
-10
-5
-5
-10
41)
20
10
-20
-10
10
20 r
-10
-20
19
Answer Key
Testname: M155_E4_PRACTICE
42)
4
3
2
1
-4
-3
-2
-1
1
2
3
4 r
1
2
3
4 r
-1
-2
-3
-4
43)
4
3
2
1
-4
-3
-2
-1
-1
-2
-3
-4
44)
3α2 π
2
45)
49π
8
46) 66π
25
47)
(2π - 3 3)
2
48)
3 3
4
49)
2
(4π - 3 3)
3
50)
25
(5π - 8)
4
51) 3(4π - 3 3)
73π
52)
4
53) 4(π + 3 3)
20
Answer Key
Testname: M155_E4_PRACTICE
54)
y
10
5
-10
-5
5
10
x
-5
-10
55)
10
y
5
-10
-5
x
5
-5
-10
56) (0, -3); y = 3
57) (7, 0); x = -7
58) B
59)
y
10
5
-10
-5
5
10
x
-5
-10
21
Answer Key
Testname: M155_E4_PRACTICE
60)
y
5
-5
x
5
-5
61) Vertices: (±11, 0); Foci: (± 105, 0)
x2
y2
62)
+
=1
49
4
63)
x2 y2
+
=1
69 81
64)
y
10
5
-10
-5
5
10
x
5
10
x
-5
-10
65)
y
10
5
-10
-5
-5
-10
66) Foci: (-10, 0), (10, 0); Asymptotes: y =
3
3
x, y = - x
4
4
67) Vertices: (0, 10), (0, -10); Asymptotes: y = ±
5
x
4
22
Answer Key
Testname: M155_E4_PRACTICE
68)
x2 y2
=1
4
5
69)
y2
x2
=1
36 100
70)
y2 x2
+
=1
9
8
71)
6
4
2
-6
-4
-2
2
4
6
2
4
6
-2
-4
-6
72)
6
4
2
-6
-4
-2
-2
-4
-6
73)
10
5
-10
-5
5
10
-5
-10
23