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Review #5 over Ch 10
Work all the problems on a separate piece of paper
showing all steps.
Identify the type of conic, any vertices, foci, directrix, asymptotes, and
also the center where applicable. Then graph the following:
1)
y2 = – 16(x – 3)
3)
+
2)
=1
–
4)
x2 + 4y2 = 16
=1
5)
16y – x2 = 0
6)
9y2 – 4x2 = 36
7)
x2 – 6x = 3y
8)
4x2 + 3y2 + 8x – 6y = 5
9)
4x2 – y2 – 24x – 4y + 16 = 0
Find the equation of the conic that satisfies the following conditions:
10) Hyperbola; focus: (– 2, – 2); vertices: (– 2, – 1) & (– 2, 3)
11) Parabola; focus: (6, – 3); vertex: (4, – 3).
12) Ellipse; center: (2, – 2); vertex: (7, – 2); focus: (4, – 2).
13) Ellipse; center: (1, 2); vertex: (1, 4); contains the point (2, 2).
14) Hyperbola; vertices: (1, – 3) & (1, 1); asymptote: y + 1 = 1.5(x – 1)
15) Parabola; Focus (– 4, 4); Directrix: y = – 2
Identify the conic:
16) y2 + 4x + 3y – 8 = 0
17)
2
2
18) 2x + 8xy + 4y – 3x + 2 = 5
19)
2
2
20) 4x – 12xy + 9y + 3x + 4y – 8 = 0
4x2 + 10y2 – x + 3y = 0
9x2 – 10xy + 7y2 + 5x – 3 = 0
Rotate the axes so there is no xy-term. Analyze and graph the equation:
21) 5x2 + 6xy + 5y2 – 8 = 0
22) 6x2 – 10 3 xy – 4y2 – 99 = 0
Identify the conic, graph it and convert the equation to rectangular form:
23) r =
24) €r =
25)
r=
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332
Find two different sets of parametric equations for the following:
26) y = 4x2 + 5
–∞<x<∞
27)
+
= 1, time is the parameter and
a) the first set must satisfy the conditions that the motion around the
ellipse is clockwise beginning at (– 3, 0) and takes 4 seconds to complete
one revolution.
b) the second set must satisfy the conditions that the motion around the
ellipse is counterclockwise beginning at (0, 2) and takes 2 seconds to
complete one revolution.
Use the given parametric equations to graph the curve and show the
orientation. Then find the rectangular equation for the curve.
28) x = 3t – 4, y = 4 – t, – ∞ < t < ∞
29)
x = ln(t), y = t3, t >
30)
x = csc(t), y = cot(t),
≤t≤
Solve the following:
31) A one-way tunnel is designed in a shape of a semielliptical arch with a
height of 16 feet and 18 feet long at the base of the arch. If an oversize
truck is 10 feet wide, how high can it be to still pass safely through the
tunnel?
32)
In making a pass in football, Tony Romo throws a football from his arm
two yards above the ground at and angle of 20˚. If the initial speed of the
ball was 25 yd/sec,
a) Find the parametric equations that describe the position of the ball
as a function of time (assume g = 32/3 yd/s2).
b) If Miles Austin catches the ball just as it is about to hit the ground, how
long was the ball in the air?
c) Find the horizontal distance the ball travelled before Miles Austin
caught the ball.
d) When is the ball at its maximum height? What is that maximum
height?
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333
Find the equation of the following that satisfies the following conditions:
33) The graph given below:
34) The graph given below:
35) The graph given below:
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Answers:
1)
Center: N/A
Vertex: (3, 0)
Focus: (– 1, 0)
Directrix: x = 7
Axis of Sym: y = 0
3)
2)
Center: (– 2, 1)
Vertices: (– 6, 1) & (2, 1)
Foci: (– 7, 1) & (3, 1)
Asym: y = – 0.75x – 0.5 or 0.75x + 2.5
Transverse Axis: y = 1
Center: (3, – 2)
4)
Vertices: (3, – 5) & (3, 1)
Foci: ≈ (3, 0. 236) & (3, – 4.236)
B-int: (5, – 2) & (1, – 2)
Major Axis: x = 3
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Center: (0, 0)
Vertices: (– 4, 0) & (4, 0)
Foci: ≈ (– 3.464, 0) & (3.464, 0)
B-int: (0, – 2) & (0, 2)
Major Axis: y = 0
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5)
Center: N/A
Vertex: (0, 0)
Focus: (0, 4)
Directrix: y = – 4
6)
Axis of Sym: x = 0
7)
Center: N/A
Vertex: (3, – 3)
Focus: ≈ (3, – 2.25)
Directrix: y = – 3.75
Axis of Sym.: x = 3
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Center: (0, 0)
Vertices: (0, – 2) & (0, 2)
Foci: ≈ (0, – 3.606) & (0, 3.606)
Asym: y = ± x
Transverse Axis: x = 0
8)
Center: (– 1, 1)
Vertices: (– 1, – 1) & (– 1, 3)
Foci: (– 1, 0) & (– 1, 2)
B-int: ≈ (– 2.732, 1) & (0.732, 1)
Major Axis: x = – 1
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9)
Center: (3, – 2)
Vertices: (1, – 2) & (5, – 2)
Foci: (– 1.472, – 2) & (7.472, – 2)
Asym: y = 2x – 8 or – 2x + 4
Transverse Axis: y = – 2
10)
11)
–
=1
(y + 3)2 = 8(x – 4)
12)
+
=1
13)
+ (x – 1)2 = 1
14)
–
= 1 15) (x + 4)2 = 12(y – 1) 16) Parabola
17) Ellipse 18) Hyperbola 19) Ellipse 20) Parabola
21) Axes Rotated by 45˚
22) Axes Rotated by 60˚
Ellipse in terms of x' & y'
Hyperbola in terms of x' & y'
Center: (0, 0)
Center: (0, 0)
Vertices: (0, ± 2)
Vertices: (0, ± 3)
Foci: (0, ±
)
Foci: (0, ± 2
)
b'-int: (± 1, 0)
Asymptotes: y' = ±
Major Axis: x' = 0
Transverse Axis: x' = 0
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x'
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23) e = 2 implies a hyperbola
Directrix || polar axis,
Directrix is 2 units above the pole
Focus at the pole
24) e = 0.5 implies an ellipse
Directrix ⊥ polar axis
Directrix is 2 units to the right of the pole
Focus at the pole
Vertices: (
Vertices: (
, 0) and (2, π)
Center: (–
, 0)
Center: (
,
,
) and (– 2,
)
)
Transverse Axis: θ =
Rect. Form:
–
Major Axis: Polar Axis
=1
25) e = 1 implies a parabola
Directrix || polar axis,
Directrix is 4 units below the pole
Focus at the pole
Vertex: (2,
)
Axis of symmetry: θ =
Rect. Form: x2 = 8(y + 2)
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Rect. Form:
+
=1
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26)
x = t, y = 4t2 + 5 or x = t3, y = 4t6 +5
27a) x = – 3cos(
t), y = 2sin(
t) 27b) x = – 3sin(πt), y = 2cos(πt)
28)
29)
y=–
x+
y = e3x, x > – 1
, –∞<x<∞
30)
31) The truck has to be most 13.3
feet high.
32a) x = 23.4923t
t2 + 8.5505t + 2
y=–
32b) It was in the air for ≈ 1.8104 sec.
32c) The ball went 42.5296 yards.
32d) Thus, the ball reached a maximum
height of about 8.8408 yd at
t ≈ 0.8016 seconds.
33)
+
(x + 3)2 = 8(y + 4)
34)
35)
{ (x, y) | y =
U { (x, y) | y = –
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,1≤x≤
,1≤x≤
=1
–
}
}
=1