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College Algebra 7th

James Stewart, Lothar Redlin, Saleem Watson

Chapter 7

Conic Sections

Educators

AG

Problem 1

A parabola is the set of all points in the plane that are equidistant from a fixed point called the _________ and a fixed line called the ____________ of the parabola.

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Problem 2

The graph of the equation $x^{2}=4 p y$ is a parabola with focus $F($ __________ , ___________ ) and directrix $y=$ So the graph of $x^{2}=12 y$ is a parabola with focus $F=$ (______________ , ________) and directrix $x=$ _____________

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Problem 3

The graph of the equation $y^{2}=4 p x$ is a parabola with focus $F($____________ , ___________ )and directrix $x=$ _____________ So the graph of $y^{2}=12 x$ is a parabola with focus $F$ ( _________ , _________ ) and directrix $x=$ ______________ .

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Problem 4

Label the focus, directrix, and vertex on the graphs given for the parabolas
a. $x^{2}=12 y$
b. $y^{2}=12 x$

AG
Ankit G.
Numerade Educator

Problem 5

Match the equation with the graphs labeled $\mathbf{I},$ II, III, $\mathrm{IV}$ V and VI. Give reasons for your answers.
$$y^{2}=2 x$$

AG
Ankit G.
Numerade Educator

Problem 6

Match the equation with the graphs labeled $\mathbf{I},$ II, III, $\mathrm{IV}$ V and VI. Give reasons for your answers.
$$y^{2}=-\frac{1}{4} x$$

AG
Ankit G.
Numerade Educator

Problem 7

Match the equation with the graphs labeled $\mathbf{I},$ II, III, $\mathrm{IV}$ V and VI. Give reasons for your answers.
$$x^{2}=-6 y$$

AG
Ankit G.
Numerade Educator

Problem 8

Match the equation with the graphs labeled $\mathbf{I},$ II, III, $\mathrm{IV}$ V and VI. Give reasons for your answers.
$$2 x^{2}=y$$

AG
Ankit G.
Numerade Educator

Problem 9

Match the equation with the graphs labeled $\mathbf{I},$ II, III, $\mathrm{IV}$ V and VI. Give reasons for your answers.
$$y^{2}-8 x=0$$

AG
Ankit G.
Numerade Educator

Problem 10

Match the equation with the graphs labeled $\mathbf{I},$ II, III, $\mathrm{IV}$ V and VI. Give reasons for your answers.
$$12 y+x^{2}=0$$

AG
Ankit G.
Numerade Educator

Problem 11

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$x^{2}=8 y$$

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Problem 12

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$x^{2}=-4 y$$

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Problem 13

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$y^{2}=-24 x$$

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Problem 14

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$y^{2}=16 x$$

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Problem 15

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$y=-\frac{1}{8} x^{2}$$

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Problem 16

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$x=2 y^{2}$$

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Problem 17

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$x=-2 y^{2}$$

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Problem 18

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$y=\frac{1}{4} x^{2}$$

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Problem 19

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$5 y=x^{2}$$

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Problem 20

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$9 x=y^{2}$$

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Problem 21

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$x^{2}+12 y=0$$

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Problem 22

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$x+\frac{1}{5} y^{2}=0$$

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Problem 23

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$5 x+3 y^{2}=0$$

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Problem 24

An equation of a parabola is given.
(a) Find the focus, directrix, and focal diameter of the parabola.
(b) Sketch a graph of the parabola and its directrix.
$$8 x^{2}+12 y=0$$

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Problem 25

Use a graphing device to graph the parabola.
$$x^{2}=16 y$$

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Problem 26

Use a graphing device to graph the parabola.
$$x^{2}=-8 y$$

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Problem 27

Use a graphing device to graph the parabola.
$$y^{2}=-\frac{1}{3} x$$

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Problem 28

Use a graphing device to graph the parabola.
$$8 y^{2}=x$$

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Problem 29

Use a graphing device to graph the parabola.
$$4 x+y^{2}=0$$

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Problem 30

Use a graphing device to graph the parabola.
$$x-2 y^{2}=0$$

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Problem 31

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Focus: $F(0,6)$

AG
Ankit G.
Numerade Educator

Problem 32

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Focus: $F\left(0,-\frac{1}{4}\right)$

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Problem 33

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Focus: $F(-8,0)$

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Problem 34

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Focus: $F(5,0)$

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Problem 35

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Focus: $F\left(0,-\frac{3}{4}\right)$

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Problem 36

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Focus: $F\left(-\frac{1}{12}, 0\right)$

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Problem 37

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Directrix: $x=-4$

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Problem 38

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Directrix: $y=\frac{1}{2}$

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Problem 39

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Directrix: $y=\frac{1}{10}$

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Problem 40

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Directrix: $x=-\frac{1}{8}$

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Problem 41

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Directrix: $x=\frac{1}{20}$

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Problem 42

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Directrix: $y=-5$

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Problem 43

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Focus on the positive $x$ -axis, 2 units away from the directrix

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Problem 44

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Focus on the negative $y$ -axis, 6 units away from the directrix

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Problem 45

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Opens downward with focus 10 units away from the vertex

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Problem 46

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Opens upward with focus 5 units away from the vertex

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Problem 47

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Directrix has $y$ -intercept 6

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Problem 48

Find an equation for the parabola that has its vertex at the origin and satisfies the given
condition(s).
Focal diameter 8 and focus on the negative $y$ -axis

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Problem 49

Find an equation of the parabola whose graph is shown.

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Problem 50

Find an equation of the parabola whose graph is shown.

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Problem 51

Find an equation of the parabola whose graph is shown.

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Problem 52

Find an equation of the parabola whose graph is shown.

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Problem 53

Find an equation of the parabola whose graph is shown.

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Problem 54

Find an equation of the parabola whose graph is shown.

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Problem 55

Find an equation of the parabola whose graph is shown.

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Problem 56

Find an equation of the parabola whose graph is shown.

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Problem 57

Find an equation of the parabola whose graph is shown.

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Problem 58

Find an equation of the parabola whose graph is shown.

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Problem 59

(a) Find equations for the family of parabolas with the given description.
(b) Draw the graphs.
What do you conclude?
The family of parabolas with vertex at the origin and with directrixes $y=\frac{1}{2}, y=1, y=4,$ and $y=8$

AG
Ankit G.
Numerade Educator

Problem 60

Find an equation of the parabola whose graph is shown.
(a) Find equations for the family of parabolas with the given description.
(b) Draw the graphs.
What do you conclude?
The family of parabolas with vertex at the origin, focus on the positive $y$ -axis, and with focal diameters $1,2,4,$ and 8

AG
Ankit G.
Numerade Educator

Problem 61

Parabolic Reflector A lamp with a parabolic reflector is shown in the figure. The bulb is placed at the focus, and the focal diameter is $12 \mathrm{cm}$.
a. Find an equation of the parabola.
b. Find the diameter $d(C, D)$ of the opening, $20 \mathrm{cm}$ from the vertex.

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Problem 62

Satellite Dish A reflector for a satellite dish is parabolic in cross section, with the receiver at the focus $F$. The reflector is $1 \mathrm{ft}$ deep and $20 \mathrm{ft}$ wide from rim to rim (See the figure). How far is the receiver from the vertex of the parabolic reflector?

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Problem 63

Suspension Bridge In a suspension bridge the shape of the suspension cables is par The bridge shown in the figure has towers that are 600 m apart, and the lowest point of the suspension cables is 150 m below the top of the towers. Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the vertex. [Note: This equation is used to find the length of cable needed in the construction of the bridge.]

AG
Ankit G.
Numerade Educator

Problem 64

Reflecting Telescope The Hale telescope at the Mount Palomar Observatory has a 200 -in. mirror, as shown in the figure. The mirror is constructed in a parabolic shape that collects light from the stars and focuses it at the prime focus, that is, the focus of the parabola. The mirror is 3.79 in. deep at its center. Find the focal length of this parabolic mirror, that is, the distance from the vertex to the focus.

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Problem 65

Discuss - Write: Parabolas in the Real World Several examples of the uses of
parabolas are given in the text. Find other situations in real life in which parabolas occur. Consult a scientific encyclopedia in the reference section of your library, or search the Internet.

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Problem 66

Discuss: Light Cone from a Flashlight A flashlight is held to form a lighted area on the ground, as shown in the figure. Is it possible to angle the flashlight in such a way that the boundary of the lighted area is a parabola? Explain your answer.

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