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Calculus of a Single Variable

Ron Larson, Bruce Edwards

Chapter 10

Conics, Parametric Equations, and Polar Coordinates

Educators


Problem 1

In Exercises $1-8,$ match the equation with its graph. [The graphs are labeled $(\mathbf{a}),(\mathbf{b}),(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),(\mathbf{f}), \mathbf{a n d}(\mathbf{h})$.] Graph cannot copy
$$
y^{2}=4 x
$$

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Problem 2

In Exercises $1-8,$ match the equation with its graph. [The graphs are labeled $(\mathbf{a}),(\mathbf{b}),(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),(\mathbf{f}), \mathbf{a n d}(\mathbf{h})$.] Graph cannot copy
$$
(x+4)^{2}=2(y+2)
$$

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Problem 3

In Exercises $1-8,$ match the equation with its graph. [The graphs are labeled $(\mathbf{a}),(\mathbf{b}),(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),(\mathbf{f}), \mathbf{a n d}(\mathbf{h})$.] Graph cannot copy
$$
(x+4)^{2}=-2(y-2)
$$

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Problem 4

In Exercises $1-8,$ match the equation with its graph. [The graphs are labeled $(\mathbf{a}),(\mathbf{b}),(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),(\mathbf{f}), \mathbf{a n d}(\mathbf{h})$.] Graph cannot copy
$$
\frac{(x-2)^{2}}{16}+\frac{(y+1)^{2}}{4}=1
$$

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Problem 5

In Exercises $1-8,$ match the equation with its graph. [The graphs are labeled $(\mathbf{a}),(\mathbf{b}),(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),(\mathbf{f}), \mathbf{a n d}(\mathbf{h})$.] Graph cannot copy
$$
\frac{x^{2}}{4}+\frac{y^{2}}{9}=1
$$

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Problem 6

In Exercises $1-8,$ match the equation with its graph. [The graphs are labeled $(\mathbf{a}),(\mathbf{b}),(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),(\mathbf{f}), \mathbf{a n d}(\mathbf{h})$.] Graph cannot copy
$$
\frac{x^{2}}{16}+\frac{y^{2}}{16}=1
$$

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Problem 7

In Exercises $1-8,$ match the equation with its graph. [The graphs are labeled $(\mathbf{a}),(\mathbf{b}),(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),(\mathbf{f}), \mathbf{a n d}(\mathbf{h})$.] Graph cannot copy
$$
\frac{y^{2}}{16}-\frac{x^{2}}{1}=1
$$

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Problem 8

In Exercises $1-8,$ match the equation with its graph. [The graphs are labeled $(\mathbf{a}),(\mathbf{b}),(\mathbf{c}),(\mathbf{d}),(\mathbf{e}),(\mathbf{f}), \mathbf{a n d}(\mathbf{h})$.] Graph cannot copy
$$
\frac{(x-2)^{2}}{9}-\frac{y^{2}}{4}=1
$$

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Problem 9

In Exercises $9-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
y^{2}=-8 x
$$

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Problem 10

In Exercises $9-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
x^{2}+6 y=0
$$

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Problem 11

In Exercises $9-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
(x+5)+(y-3)^{2}=0
$$

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Problem 12

In Exercises $9-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
(x-6)^{2}+8(y+7)=0
$$

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Problem 13

In Exercises $9-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
y^{2}-4 y-4 x=0
$$

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Problem 14

In Exercises $9-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
y^{2}+6 y+8 x+25=0
$$

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Problem 15

In Exercises $9-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
x^{2}+4 x+4 y-4=0
$$

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Problem 16

In Exercises $9-16,$ find the vertex, focus, and directrix of the parabola, and sketch its graph.
$$
y^{2}+4 y+8 x-12=0
$$

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Problem 17

In Exercises $17-20$ , find the vertex, focus, and directrix of the parabola. Then use a graphing utility to graph the parabola.
$$
y^{2}+x+y=0
$$

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Problem 18

In Exercises $17-20$ , find the vertex, focus, and directrix of the parabola. Then use a graphing utility to graph the parabola.
$$
y=-\frac{1}{6}\left(x^{2}-8 x+6\right)
$$

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Problem 19

In Exercises $17-20$ , find the vertex, focus, and directrix of the parabola. Then use a graphing utility to graph the parabola.
$$
y^{2}-4 x-4=0
$$

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Problem 20

In Exercises $17-20$ , find the vertex, focus, and directrix of the parabola. Then use a graphing utility to graph the parabola.
$$
x^{2}-2 x+8 y+9=0
$$

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Problem 21

In Exercises $21-28,$ find an equation of the parabola.
$$
\begin{array}{l}{\text { Vertex: }(5,4)} \\ {\text { Focus: }(3,4)}\end{array}
$$

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Problem 22

In Exercises $21-28,$ find an equation of the parabola.
$$
\begin{array}{l}{\text { Vertex: }(-2,1)} \\ {\text { Focus: }(-2,-1)}\end{array}
$$

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Problem 23

In Exercises $21-28,$ find an equation of the parabola.
$$
\begin{array}{l}{\text { Vertex: }(0,5)} \\ {\text { Directrix: } y=-3}\end{array}
$$

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Problem 24

In Exercises $21-28,$ find an equation of the parabola.
$$
\begin{array}{l}{\text { Focus: }(2,2)} \\ {\text { Directrix: } x=-2}\end{array}
$$

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Problem 25

In Exercises $21-28,$ find an equation of the parabola.
Graph cannot copy

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Problem 26

In Exercises $21-28,$ find an equation of the parabola.
Graph cannot copy

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Problem 27

In Exercises $21-28,$ find an equation of the parabola.
Axis is parallel to $y$ -axis; graph passes through $(0,3),(3,4),$ and $(4,11) .$

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Problem 28

In Exercises $21-28,$ find an equation of the parabola.
Directrix: $y=-2 ;$ endpoints of latus rectum are $(0,2)$ and $(8,2) .$

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Problem 29

In Exercises $29-34,$ find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
16 x^{2}+y^{2}=16
$$

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Problem 30

In Exercises $29-34,$ find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
3 x^{2}+7 y^{2}=63
$$

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Problem 31

In Exercises $29-34,$ find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
\frac{(x-3)^{2}}{16}+\frac{(y-1)^{2}}{25}=1
$$

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Problem 32

In Exercises $29-34,$ find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
(x+4)^{2}+\frac{(y+6)^{2}}{1 / 4}=1
$$

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Problem 33

In Exercises $29-34,$ find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
9 x^{2}+4 y^{2}+36 x-24 y+36=0
$$

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Problem 34

In Exercises $29-34,$ find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.
$$
16 x^{2}+25 y^{2}-64 x+150 y+279=0
$$

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Problem 35

In Exercises $35-38$ , find the center, foci, and vertices of the ellipse. Use a graphing utility to graph the ellipse.
$$
12 x^{2}+20 y^{2}-12 x+40 y-37=0
$$

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Problem 36

In Exercises $35-38$ , find the center, foci, and vertices of the ellipse. Use a graphing utility to graph the ellipse.
$$
36 x^{2}+9 y^{2}+48 x-36 y+43=0
$$

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Problem 37

In Exercises $35-38$ , find the center, foci, and vertices of the ellipse. Use a graphing utility to graph the ellipse.
$$
x^{2}+2 y^{2}-3 x+4 y+0.25=0
$$

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Problem 38

In Exercises $35-38$ , find the center, foci, and vertices of the ellipse. Use a graphing utility to graph the ellipse.
$$
2 x^{2}+y^{2}+4.8 x-6.4 y+3.12=0
$$

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Problem 39

In Exercises $39-44,$ find an equation of the ellipse.
$$
\begin{array}{l}{\text { Center: }(0,0)} \\ {\text { Focus: }(5,0)} \\ {\text { Vertex: }(6,0)}\end{array}
$$

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Problem 40

In Exercises $39-44,$ find an equation of the ellipse.
$$
\begin{array}{l}{\text { Vertices: }(0,3),(8,3)} \\ {\text { Eccentricity: } \frac{3}{4}}\end{array}
$$

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Problem 41

In Exercises $39-44,$ find an equation of the ellipse.
$$
\begin{array}{l}{\text { Vertices: }(3,1),(3,9)} \\ {\text { Minor axis length: } 6}\end{array}
$$

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Problem 42

In Exercises $39-44,$ find an equation of the ellipse.
$$
\begin{array}{l}{\text { Foci: }(0, \pm 9)} \\ {\text { Major axis length: } 22}\end{array}
$$

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Problem 43

In Exercises $39-44,$ find an equation of the ellipse.
$$
\begin{array}{l}{\text { Center: }(0,0)} \\ {\text { Major axis: horizontal }} \\ {\text { Points on the ellipse: }} \\ {(3,1),(4,0)}\end{array}
$$

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Problem 44

In Exercises $39-44,$ find an equation of the ellipse.
$$
\begin{array}{l}{\text { Center: }(1,2)} \\ {\text { Major axis: vertical }} \\ {\text { Points on the ellipse: }} \\ {(1,6),(3,2)}\end{array}
$$

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Problem 45

In Exercises $45-52$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
y^{2}-\frac{x^{2}}{9}=1
$$

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Problem 46

In Exercises $45-52$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
\frac{x^{2}}{25}-\frac{y^{2}}{16}=1
$$

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Problem 47

In Exercises $45-52$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
\frac{(x-1)^{2}}{4}-\frac{(y+2)^{2}}{1}=1
$$

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Problem 48

In Exercises $45-52$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
\frac{(y+3)^{2}}{225}-\frac{(x-5)^{2}}{64}=1
$$

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Problem 49

In Exercises $45-52$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
9 x^{2}-y^{2}-36 x-6 y+18=0
$$

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Problem 50

In Exercises $45-52$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
y^{2}-16 x^{2}+64 x-208=0
$$

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Problem 51

In Exercises $45-52$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
x^{2}-9 y^{2}+2 x-54 y-80=0
$$

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Problem 52

In Exercises $45-52$ , find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.
$$
9 x^{2}-4 y^{2}+54 x+8 y+78=0
$$

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Problem 53

In Exercises $53-56,$ find the center, foci, and vertices of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes.
$$
9 y^{2}-x^{2}+2 x+54 y+62=0
$$

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Problem 54

In Exercises $53-56,$ find the center, foci, and vertices of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes.
$$
9 x^{2}-y^{2}+54 x+10 y+55=0
$$

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Problem 55

In Exercises $53-56,$ find the center, foci, and vertices of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes.
$$
3 x^{2}-2 y^{2}-6 x-12 y-27=0
$$

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Problem 56

In Exercises $53-56,$ find the center, foci, and vertices of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes.
$$
3 y^{2}-x^{2}+6 x-12 y=0
$$

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Problem 57

In Exercises $57-64,$ find an equation of the hyperbola.
$$
\begin{array}{l}{\text { Vertices: }( \pm 1,0)} \\ {\text { Asymptotes: } y=\pm 5 x}\end{array}
$$

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Problem 58

In Exercises $57-64,$ find an equation of the hyperbola.
$$
\begin{array}{l}{\text { Vertices: }(0, \pm 4)} \\ {\text { Asymptotes: } y=\pm 2 x}\end{array}
$$

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Problem 59

In Exercises $57-64,$ find an equation of the hyperbola.
$$
\begin{array}{l}{\text { Vertices: }(2, \pm 3)} \\ {\text { Point on graph: }(0,5)}\end{array}
$$

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Problem 60

In Exercises $57-64,$ find an equation of the hyperbola.
$$
\begin{array}{l}{\text { Vertices: }(2, \pm 3)} \\ {\text { Foci: }(2, \pm 5)}\end{array}
$$

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Problem 61

In Exercises $57-64,$ find an equation of the hyperbola.
$$
\begin{array}{l}{\text { Center: }(0,0)} \\ {\text { Vertex: }(0,2)} \\ {\text { Focus: }(0,4)}\end{array}
$$

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Problem 62

In Exercises $57-64,$ find an equation of the hyperbola.
$$
\begin{array}{l}{\text { Center: }(0,0)} \\ {\text { Vertex: }(6,0)} \\ {\text { Focus: }(10,0)}\end{array}
$$

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Problem 63

In Exercises $57-64,$ find an equation of the hyperbola.
$$
\begin{array}{l}{\text { Vertices: }(0,2),(6,2)} \\ {\text { Asymptotes: } y=\frac{2}{3} x} \\ {y=4-\frac{2}{3} x}\end{array}
$$

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Problem 64

In Exercises $57-64,$ find an equation of the hyperbola.
$$
\begin{array}{l}{\text { Focus: }(20,0)} \\ {\text { Asymptotes: } y=\pm \frac{3}{4} x}\end{array}
$$

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Problem 65

In Exercises 65 and $66,$ find equations for (a) the tangent lines and (b) the normal lines to the hyperbola for the given value of $x .$
$$
\frac{x^{2}}{9}-y^{2}=1, \quad x=6
$$

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Problem 66

In Exercises 65 and $66,$ find equations for (a) the tangent lines and (b) the normal lines to the hyperbola for the given value of $x .$
$$
\frac{y^{2}}{4}-\frac{x^{2}}{2}=1, \quad x=4
$$

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Problem 67

In Exercises $67-76,$ classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
x^{2}+4 y^{2}-6 x+16 y+21=0
$$

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Problem 68

In Exercises $67-76,$ classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
4 x^{2}-y^{2}-4 x-3=0
$$

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Problem 69

In Exercises $67-76,$ classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
y^{2}-8 y-8 x=0
$$

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Problem 70

In Exercises $67-76,$ classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
25 x^{2}-10 x-200 y-119=0
$$

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Problem 71

In Exercises $67-76,$ classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
4 x^{2}+4 y^{2}-16 y+15=0
$$

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Problem 72

In Exercises $67-76,$ classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
y^{2}-4 y=x+5
$$

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Problem 73

In Exercises $67-76,$ classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
9 x^{2}+9 y^{2}-36 x+6 y+34=0
$$

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Problem 74

In Exercises $67-76,$ classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
2 x(x-y)=y(3-y-2 x)
$$

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Problem 75

In Exercises $67-76,$ classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
3(x-1)^{2}=6+2(y+1)^{2}
$$

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Problem 76

In Exercises $67-76,$ classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
$$
9(x+3)^{2}=36-4(y-2)^{2}
$$

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Problem 77

(a) Give the definition of a parabola.
(b) Give the standard forms of a parabola with vertex at $(h, k) .$
(c) In your own words, state the reflective property of a parabola.

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Problem 78

(a) Give the definition of an ellipse.
(b) Give the standard forms of an ellipse with center at $(h, k)$

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Problem 79

(a) Give the definition of a hyperbola.
(b) Give the standard forms of a hyperbola with center at $(h, k) .$
(c) Write equations for the asymptotes of a hyperbola.

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Problem 80

Define the eccentricity of an ellipse. In your own words, describe how changes in the eccentricity affect the ellipse.

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Problem 81

Solar Collector $A$ solar collector for heating water is constructed with a sheet of stainless steel that is formed into the shape of a parabola (see figure). The water will flow through a pipe that is located at the focus of the parabola. At what distance from the vertex is the pipe?

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Problem 82

Beam Deflection A simply supported beam that is 16 meters long has a load concentrated at the center (see figure). The deflection of the beam at its center is 3 centimeters. Assume that the shape of the deflected beam is parabolic.
(a) Find an equation of the parabola. (Assume that the origin is at the center of the beam.)
(b) How far from the center of the beam is the deflection 1 centimeter?

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Problem 83

Find an equation of the tangent line to the parabola $y=a x^{2}$ at $x=x_{0} .$ Prove that the $x$ -intercept of this tangent line is $\left(x_{0} / 2,0\right) .$

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Problem 84

(a) Prove that any two distinct tangent lines to a parabola intersect.
(b) Demonstrate the result of part (a) by finding the point of intersection of the tangent lines to the parabola $x^{2}-4 x-4 y=0$ at the points $(0,0)$ and $(6,3) .$

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Problem 85

(a) Prove that if any two tangent lines to a parabola intersect at right angles, their point of intersection must lie on the directrix.
(b) Demonstrate the result of part (a) by proving that the tangent lines to the parabola $x^{2}-4 x-4 y+8=0$ at the points $(-2,5)$ and $\left(3, \frac{5}{4}\right)$ intersect at right angles, and that the point of intersection lies on the directrix.

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Problem 86

Find the point on the graph of $x^{2}=8 y$ that is closest to the focus of the parabola.

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Problem 87

Radio and Television Reception In mountainous areas, reception of radio and television is sometimes poor. Consider an idealized case where a hill is represented by the graph of the parabola $y=x-x^{2},$ a transmitter is located at the point $(-1,1),$ and a receiver is located on the other side of the hill at the point $\left(x_{0}, 0\right) .$ What is the closest the receiver can be to the hill while still maintaining unobstructed reception?

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Problem 88

Modeling Data The table shows the average amounts of time $A$ (in minutes) women spent watching television each day for the years 1999 through 2005 . (Source: Nielsen Media Research)
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline \text { Year } & {1999} & {2000} & {2001} & {2002} & {2003} & {2004} & {2005} \\ \hline A & {280} & {286} & {291} & {298} & {305} & {307} & {317} \\ \hline\end{array}
$$
(a) Use the regression capabilities of a graphing utility to find a model of the form $A=a t^{2}+b t+c$ for the data. Let $t$ represent the year, with $t=9$ corresponding to $1999 .$
(b) Use a graphing utility to plot the data and graph the model.
(c) Find $d A / d t$ and sketch its graph for $9 \leq t \leq 15 .$ What information about the average amount of time women spent watching television is given by the graph of the derivative?

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Problem 89

Architecture A church window is bounded above by a parabola and below by the arc of a circle (see figure). Find the surface area of the window.

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Problem 90

Arc Length Find the arc length of the parabola $4 x-y^{2}=0$ over the interval $0 \leq y \leq 4$.

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Problem 91

Bridge Design $A$ cable of a suspension bridge is suspended (in the shape of a parabola) between two towers that are 120 meters apart and 20 meters above the roadway (see figure). The cables touch the roadway midway between the towers.
(a) Find an equation for the parabolic shape of each cable.
(b) Find the length of the parabolic supporting cable.

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Problem 92

Surface Area A satellite signal receiving dish is formed by revolving the parabola given by $x^{2}=20 y$ about the $y$ -axis. The radius of the dish is $r$ feet. Verify that the surface area of the dish is given by
$$
2 \pi \int_{0}^{r} x \sqrt{1+\left(\frac{x}{10}\right)^{2}} d x=\frac{\pi}{15}\left[\left(100+r^{2}\right)^{3 / 2}-1000\right]
$$

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Problem 93

Investigation Sketch the graphs of $x^{2}=4 p y$ for $p=\frac{1}{4}, \frac{1}{2}, 1, \frac{3}{2},$ and 2 on the same coordinate axes. Discuss the change in the graphs as $p$ increases.

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Problem 94

Area Find a formula for the area of the shaded region in the figure. Graph cannot copy

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Problem 95

Writing On page 699 , it was noted that an ellipse can be drawn using two thumbtacks, a string of fixed length (greater than the distance between the tacks), and a pencil. If the ends of the string are fastened at the tacks and the string is drawn taut with a pencil, the path traced by the pencil will be an ellipse.
(a) What is the length of the string in terms of $a ?$
(b) Explain why the path is an ellipse.

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Problem 96

Construction of a Semielliptical Arch A fireplace arch is to be constructed in the shape of a semiellipse. The opening is to have a height of 2 feet at the center and a width of 5 feet along the base (see figure). The contractor draws the outline of the ellipse by the method shown in Exercise $95 .$ Where should the tacks be placed and what should be the length of the piece of string?

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Problem 97

Sketch the ellipse that consists of all points $(x, y)$ such that the sum of the distances between $(x, y)$ and two fixed points is 16 units, and the foci are located at the centers of the two sets of concentric circles in the figure. To print an enlarged copy of the graph, go to the website www. mathgraphs.com.

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Problem 98

Orbit of Earth Earth moves in an elliptical orbit with the sun at one of the foci. The length of half of the major axis is $149,598,000$ kilometers, and the eccentricity is $0.0167 .$ Find the minimum distance (perihelion) and the maximum distance (aphelion) of Earth from the sun.

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Problem 99

Satellite Orbit The apogee (the point in orbit farthest from Earth) and the perigee (the point in orbit closest to Earth) of an elliptical orbit of an Earth satellite are given by $A$ and $P$ . Show that the eccentricity of the orbit is
$$
e=\frac{A-P}{A+P}
$$

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Problem 100

Explorer 18 On November $27,1963$ , the United States launched the research satellite Explorer $18 .$ Its low and high points above the surface of Earth were 119 miles and $123,000$ miles. Find the eccentricity of its elliptical orbit.

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Problem 101

Explorer 55 On November $20,1975$ , the United States launched the research satellite Explorer $55 .$ Its low and high points above the surface of Earth were 96 miles and 1865 miles. Find the eccentricity of its elliptical orbit.

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Problem 102

Consider the equation
$$9 x^{2}+4 y^{2}-36 x-24 y-36=0$$
(a) Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
(b) Change the 4$y^{2}$ -term in the equation to $-4 y^{2} .$ Classify the graph of the new equation.
(c) Change the 9$x^{2}$ -term in the original equation to 4$x^{2}$ . Classify the graph of the new equation.
(d) Describe one way you could change the original equation so that its graph is a parabola.

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Problem 103

Halley's Comet Probably the most famous of all comets, Halley's comet, has an elliptical orbit with the sun at one focus. Its maximum distance from the sun is approximately 35.29 $\mathrm{AU}$ (1 astronomical unit $\approx 92.956 \times 10^{6}$ miles, and its minimum distance is approximately 0.59 AU. Find the eccentricity of the orbit.

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Problem 104

The equation of an ellipse with its center at the origin can be written as
$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}\left(1-e^{2}\right)}=1$$
Show that as $e \rightarrow 0,$ with $a$ remaining fixed, the ellipse approaches a circle.

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Problem 105

Consider a particle traveling clockwise on the elliptical path
$$\frac{x^{2}}{100}+\frac{y^{2}}{25}=1$$
The particle leaves the orbit at the point $(-8,3)$ and travels in a straight line tangent to the ellipse. At what point will the particle cross the $y$ -axis?

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Problem 106

Volume The water tank on a fire truck is 16 feet long, and its cross sections are ellipses. Find the volume of water in the partially filled tank as shown in the figure. Graph cannot copy

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Problem 107

In Exercises 107 and $108,$ determine the points at which $d y / d x$ is zero or does not exist to locate the endpoints of the major and minor axes of the ellipse.
$$
16 x^{2}+9 y^{2}+96 x+36 y+36=0
$$

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Problem 108

In Exercises 107 and $108,$ determine the points at which $d y / d x$ is zero or does not exist to locate the endpoints of the major and minor axes of the ellipse.
$$
9 x^{2}+4 y^{2}+36 x-24 y+36=0
$$

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Problem 109

Area and Volume In Exercises 109 and $110,$ find (a) the area of the region bounded by the ellipse, (b) the volume and surface area of the solid generated by revolving the region about its major axis (prolate spheroid), and (c) the volume and surface area of the solid generated by revolving the region about its minor axis (oblate spheroid).
$$
\frac{x^{2}}{4}+\frac{y^{2}}{1}=1
$$

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Problem 110

Area and Volume In Exercises 109 and $110,$ find (a) the area of the region bounded by the ellipse, (b) the volume and surface area of the solid generated by revolving the region about its major axis (prolate spheroid), and (c) the volume and surface area of the solid generated by revolving the region about its minor axis (oblate spheroid).
$$
\frac{x^{2}}{16}+\frac{y^{2}}{9}=1
$$

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Problem 111

Arc Length Use the integration capabilities of a graphing utility to approximate to two-decimal-place accuracy the elliptical integral representing the circumference of the ellipse
$$
\frac{x^{2}}{25}+\frac{y^{2}}{49}=1
$$

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Problem 112

Prove Theorem 10.4 by showing that the tangent line to an ellipse at a point $P$ makes equal angles with lines through $P$ and the foci (see figure). [Hint: $(1)$ Find the slope of the tangent line at $P,(2)$ find the slopes of the lines through $P$ and each focus, and $(3)$ use the formula for the tangent of the angle between two lines.]

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Problem 113

Geometry The area of the ellipse in the figure is twice the area of the circle. What is the length of the major axis?

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Problem 114

Conjecture
(a) Show that the equation of an ellipse can be written as
$$
\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{a^{2}\left(1-e^{2}\right)}=1
$$
(b) Use a graphing utility to graph the ellipse
$$
\frac{(x-2)^{2}}{4}+\frac{(y-3)^{2}}{4\left(1-e^{2}\right)}=1
$$
$\quad$ for $e=0.95, e=0.75, e=0.5, e=0.25,$ and $e=0$
(c) Use the results of part (b) to make a conjecture about the change in the shape of the ellipse as $e$ approaches $0 .$

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Problem 115

Find an equation of the hyperbola such that for any point on the hyperbola, the difference between its distances from the points $(2,2)$ and $(10,2)$ is $6 .$

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Problem 116

Find an equation of the hyperbola such that for any point on the hyperbola, the difference between its distances from the points $(-3,0)$ and $(-3,3)$ is 2 .

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Problem 117

Sketch the hyperbola that consists of all points $(x, y)$ such that the difference of the distances between $(x, y)$ and two fixed points is 10 units, and the foci are located at the centers of the two sets of concentric circles in the figure. To print an enlarged copy of the graph, go to the website www. mathgraphs.com.

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Problem 118

Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form:
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
$$

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Problem 119

Sound Location A rifle positioned at point $(-c, 0)$ is fired at a target positioned at point $(c, 0) .$ A person hears the sound of the rifle and the sound of the bullet hitting the target at the same time. Prove that the person is positioned on one branch of the hyperbola given by
$$
\frac{x^{2}}{c^{2} v_{s}^{2} / v_{m}^{2}}-\frac{y^{2}}{c^{2}\left(v_{m}^{2}-v_{s}^{2}\right) / v_{m}^{2}}=1
$$
where $v_{m}$ is the muzzle velocity of the rifle and $v_{s}$ is the speed of sound, which is about 1100 feet per second.

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Problem 120

Navigation LORAN (long distance radio navigation) for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light $(186,000 \text { miles per second). The difference }$ in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, 300 miles apart, are positioned on a rectangular coordinate system at $(-150,0)$ and $(150,0)$ and that a ship is traveling on a path with coordinates $(x, 75)$ (see figure). Find the $x$ -coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 1000 microseconds ( 0.001 second). Graph cannot copy

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Problem 121

Hyperbolic Mirror A hyperbolic mirror (used in some telescopes) has the property that a light ray directed at the focus will be reflected to the other focus. The mirror in the figure has the equation $\left(x^{2} / 36\right)-\left(y^{2} / 64\right)=1 .$ At which point on the mirror will light from the point $(0,10)$ be reflected to the other focus?

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Problem 122

Show that the equation of the tangent line to $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ at the point $\left(x_{0}, y_{0}\right)$ is $\left(x_{0} / a^{2}\right) x-\left(y_{0} / b^{2}\right) y=1$.

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Problem 123

Show that the graphs of the equations intersect at right angles: $\frac{x^{2}}{a^{2}}+\frac{2 y^{2}}{b^{2}}=1 \quad$ and $\quad \frac{x^{2}}{a^{2}-b^{2}}-\frac{2 y^{2}}{b^{2}}=1$

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Problem 124

Prove that the graph of the equation
$$A x^{2}+C y^{2}+D x+E y+F=0$$
is one of the following (except in degenerate cases).
$$
\text{Conic} \quad \text{Condition}
$$
$$
\begin{array}{ll}{\text { (a) Circle }} & {A=C} \\ {\text { (b) Parabola }} & {A=0 \text { or } C=0 \text { (but not both) }} \\ {\text { (c) Ellipse }} & {A C>0} \\ {\text { (d) Hyperbola }} & {A C<0}\end{array}
$$

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Problem 125

True or False? In Exercises $125-130$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
It is possible for a parabola to intersect its directrix.

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Problem 126

True or False? In Exercises $125-130$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
The point on a parabola closest to its focus is its vertex.

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Problem 127

True or False? In Exercises $125-130$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $C$ is the circumference of the ellipse
$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, \quad b<a$$
then $2 \pi b \leq C \leq 2 \pi a$

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Problem 128

True or False? In Exercises $125-130$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $D \neq 0$ or $E \neq 0,$ then the graph of $y^{2}-x^{2}+D x+E y=0$ is a hyperbola.

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Problem 129

True or False? In Exercises $125-130$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If the asymptotes of the hyperbola $\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1$ intersect at right angles, then $a=b$ .

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Problem 130

True or False? In Exercises $125-130$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
Every tangent line to a hyperbola intersects the hyperbola only at the point of tangency.

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Problem 131

For a point $P$ on an ellipse, let $d$ be the distance from the center of the ellipse to the line tangent to the ellipse at $P .$ Prove that $\left(P F_{1}\right)\left(P F_{2}\right) d^{2}$ is constant as $P$ varies on the ellipse, where $P F_{1}$ and $P F_{2}$ are the distances from $P$ to the foci $F_{1}$ and $F_{2}$ of the ellipse.

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Problem 132

Find the minimum value of $(u-v)^{2}+\left(\sqrt{2-u^{2}}-\frac{9}{v}\right)^{2}$ for $0<u<\sqrt{2}$ and $v>0$.

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