College Algebra

R. David Gustafson, Jeffrey D. Hughes

Chapter 7

Conic Sections and Quadratic Systems - all with Video Answers

Educators

Section 1

The Circle and the Parabola

Problem 1

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises.
$(x-2)^{2}+(y+5)^{2}=9:$ center $(__,__)$
radius__.

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 2

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises.
$$x^{2}+y^{2}-36=0 \text { : center }(__,__) ; \text { radius__ }$$

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 3

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises.
$x^{2}+y^{2}=5:$ center $(__,__) ;$ radius___

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 4

You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises.
$2(x-9)^{2}+2 y^{2}=7:$ center $(__,__) ;$ radius___

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 5

Determine whether the parabolic graph of the equation opens upward, downward, to the left, or to the right.
$$y^{2}=-4 x \text { ; opens }$$

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 6

Determine whether the parabolic graph of the equation opens upward, downward, to the left, or to the right.
$$y^{2}=10 x: \text { opens }$$_____

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 7

Determine whether the parabolic graph of the equation opens upward, downward, to the left, or to the right.
$x^{2}=-8(y-3)$
opens_______

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 8

Determine whether the parabolic graph of the equation opens upward, downward, to the left, or to the right.
$(x-2)^{2}=(y+3)$
opens_______

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 9

Fill in the blanks.
A parabola is the set of all points in a plane equidistant from a line, called the____ and a fixed point not on the line, called the____.

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 10

Fill in the blanks.
The general form of a second-degree equation in the variables $x$ and $y$ is
$A x^{2}$______ $=0$

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 11

Identify the conic as a circle or parabola.
$$x^{2}-5 x+y^{2}=12$$

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 12

Identify the conic as a circle or parabola.
$$3 x^{2}+3 y^{2}+18 x+6 y=24$$

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 13

Identify the conic as a circle or parabola.
$$x^{2}-8 y=6 x-1$$

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 14

Identify the conic as a circle or parabola.
$$2 y^{2}+4 y-6 x=4$$

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 15

Write an equation of each circle shown or described. Write your answer in standard form and then general form.
(FIGURE CAN'T COPY)

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 16

Write an equation of each circle shown or described. Write your answer in standard form and then general form.
(FIGURE CAN'T COPY)

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 17

Write an equation of each circle shown or described. Write your answer in standard form and then general form.
(FIGURE CAN'T COPY)

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 18

Write an equation of each circle shown or described. Write your answer in standard form and then general form.
(FIGURE CAN'T COPY)

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 19

Write an equation of each circle shown or described. Write your answer in standard form and then general form.

Radius of $6 ;$ center at the intersection of $3 x+y=1$ and $-2 x-3 y=4$

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 20

Write an equation of each circle shown or described. Write your answer in standard form and then general form.
Radius of $8 ;$ center at the intersection of $x+2 y=8$ and $2 x-3 y=-5$

Ernest Castorena

Ernest Castorena

Numerade Educator

Problem 21

Graph each circle.
(GRAPH CAN'T COPY)
$$x^{2}+y^{2}=4$$

Kirsten Bauck

Numerade Educator

Problem 22

Graph each circle.
(GRAPH CAN'T COPY)
$$x^{2}-2 x+y^{2}=15$$

Kirsten Bauck

Numerade Educator

Problem 23

Graph each circle.
(GRAPH CAN'T COPY)
$$3 x^{2}+3 y^{2}-12 x-6 y=12$$

Kirsten Bauck

Numerade Educator

Problem 24

Graph each circle.
(GRAPH CAN'T COPY)
$$2 x^{2}+2 y^{2}+4 x-8 y+2=0$$

Kirsten Bauck

Numerade Educator

Problem 25

Find the vertex, focus, and directrix of each parabola.
$$x^{2}=12 y$$

Kirsten Bauck

Numerade Educator

Problem 26

Find the vertex, focus, and directrix of each parabola.
$$y^{2}=-12 y$$

Kirsten Bauck

Numerade Educator

Problem 27

Find the vertex, focus, and directrix of each parabola.
$$(y-3)^{2}=20 x$$

Kirsten Bauck

Numerade Educator

Problem 28

Find the vertex, focus, and directrix of each parabola.
$$x^{2}=-\frac{1}{2}(y+5)$$

Kirsten Bauck

Numerade Educator

Problem 29

Find the vertex, focus, and directrix of each parabola.
$$(x+2)^{2}=-24(y-1)$$

Kirsten Bauck

Numerade Educator

Problem 30

Find the vertex, focus, and directrix of each parabola.
$$(y+1)^{2}=28(x-2)$$

Kirsten Bauck

Numerade Educator

Problem 31

Find an equation in standard form of each parabola described.
Vertex at $(0,0) ;$ focus at $(0,3)$

Kirsten Bauck

Numerade Educator

Problem 32

Find an equation in standard form of each parabola described.

Vertex at $(0,0) ;$ focus at $(0,-3)$

Kirsten Bauck

Numerade Educator

Problem 33

Find an equation in standard form of each parabola described.

Vertex at $(0,0) ;$ focus at $(-3,0)$

Kirsten Bauck

Numerade Educator

Problem 34

Find an equation in standard form of each parabola described.

Vertex at $(0,0) ;$ focus at $(3,0)$

Kirsten Bauck

Numerade Educator

Problem 35

Find an equation in standard form of each parabola described.

Vertex at $(3,5) ;$ focus at $(3,2)$

Kirsten Bauck

Numerade Educator

Problem 36

Find an equation in standard form of each parabola described.

Vertex at $(3,5) ;$ focus at $(-3,5)$

Kirsten Bauck

Numerade Educator

Problem 37

Find an equation in standard form of each parabola described.

Vertex at $(3,5) ;$ focus at $(3,-2)$

Kirsten Bauck

Numerade Educator

Problem 38

Find an equation in standard form of each parabola described.

Vertex at $(3,5)$; focus at $(6,5)$

Kirsten Bauck

Numerade Educator

Problem 39

Find an equation in standard form of each parabola described.

Vertex at $(0,2) ;$ directrix at $y=3$

Kirsten Bauck

Numerade Educator

Problem 40

Find an equation in standard form of each parabola described.

Vertex at $(-3,4) ;$ directrix at $y=2$

Kirsten Bauck

Numerade Educator

Problem 41

Find an equation in standard form of each parabola described.

Vertex at $(1,-5) ;$ directrix at $x=-1$

Kirsten Bauck

Numerade Educator

Problem 42

Find an equation in standard form of each parabola described.

Vertex at $(3,5) ;$ directrix at $x=6$

Kirsten Bauck

Numerade Educator

Problem 43

Find an equation in standard form of each parabola described.

Vertex at $(2,2) ;$ passes through $(0,0)$

Kirsten Bauck

Numerade Educator

Problem 44

Find an equation in standard form of each parabola described.

Vertex at $(-2,-2) ;$ passes through $(0,0)$

Kirsten Bauck

Numerade Educator

Problem 45

Find an equation in standard form of each parabola described.

Vertex at $(-4,6) ;$ passes through $(0,3)$

Kirsten Bauck

Numerade Educator

Problem 46

Find an equation in standard form of each parabola described.

Vertex at $(-2,3) ;$ passes through $(0,-3)$

Kirsten Bauck

Numerade Educator

Problem 47

Find an equation in standard form of each parabola described.

Vertex at $(6,8) ;$ passes through $(5,10)$ and $(5,6)$

Kirsten Bauck

Numerade Educator

Problem 48

Find an equation in standard form of each parabola described.

Vertex at $(2,3) ;$ passes through $\left(1, \frac{13}{4}\right)$ and $\left(-1, \frac{21}{4}\right)$

Kirsten Bauck

Numerade Educator

Problem 49

Find an equation in standard form of each parabola described.

Vertex at $(3,1) ;$ passes through $(4,3)$ and $(2,3)$

Kirsten Bauck

Numerade Educator

Problem 50

Find an equation in standard form of each parabola described.

Vertex at $(-4,-2) ;$ passes through $(-3,0)$ and $\left(\frac{9}{4}, 3\right)$

Kirsten Bauck

Numerade Educator

Problem 51

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$y=x^{2}+4 x+5$$

Kirsten Bauck

Numerade Educator

Problem 52

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$2 x^{2}-12 x-7 y=10$$

Kirsten Bauck

Numerade Educator

Problem 53

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$y^{2}+4 x-6 y=-1$$

Kirsten Bauck

Numerade Educator

Problem 54

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$x^{2}-2 y-2 x=-7$$

Kirsten Bauck

Numerade Educator

Problem 55

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$y^{2}-4 y=4 x-8$$

Kirsten Bauck

Numerade Educator

Problem 56

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$y^{2}+2 x-2 y=5$$

Kirsten Bauck

Numerade Educator

Problem 57

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$y^{2}-4 y=-8 x+20$$

Kirsten Bauck

Numerade Educator

Problem 58

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$y^{2}-2 y=9 x+17$$

Kirsten Bauck

Numerade Educator

Problem 59

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$x^{2}-6 y+22=-4 x$$

Kirsten Bauck

Numerade Educator

Problem 60

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$4 y^{2}-4 y+16 x=7$$

Kirsten Bauck

Numerade Educator

Problem 61

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$4 x^{2}-4 x+32 y=47$$

Kirsten Bauck

Numerade Educator

Problem 62

Write each parabola in standard form and graph it.
(GRAPH CAN'T COPY)
$$4 y^{2}-16 x+17=20 y$$

Kirsten Bauck

Numerade Educator

Problem 63

Use a graphing calculator to graph the parabola $y^{2}=4 x-12 .$ Sketch the parabola by hand and compare the results.

Kirsten Bauck

Numerade Educator

Problem 64

Use a graphing calculator to graph the parabola $y^{2}+8 x-24=0 .$ Sketch the parabola by hand and compare the results.

Kirsten Bauck

Numerade Educator

Problem 65

Broadcast range $A$ television tower broadcasts a signal with a circular range, as shown in the illustration. Can a city 50 miles east and 70 miles north of the tower receive the signal?
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 66

Warning sirens A tornado warning siren can be heard in the circular range shown in the illustration. Can a person 4 miles west and 5 miles south of the eiren hear its sound?
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 67

Radio translators Some radio stations extend their broadcast range by installing a translator $-\mathbf{a}$ remote device that receives the signal and retransmits it. A station with a broadcast range given by $x^{2}+y^{2}=1,600,$ where $x$ and $y$ are in miles, installs a translator with a broadcast area bounded by $x^{2}+y^{2}-70 y+600=0 .$ Find the greatest distance from the main transmitter that the signal can be received.

Kirsten Bauck

Numerade Educator

Problem 68

Ripples in a pond When a stone is thrown into the center of a pond, the ripples spread out in a circular pattern, moving at a rate of 3 feet per second. If the stone is dropped at the point $(0,0)$ in the illustration, when will the ripple reach the seagull floating at the point $(15,36) ?$
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 69

Writing equations of circles Find an cquation in standard form of the circle whose outer rim
is the circular arch shown in the illustration.
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 70

Writing equations of circles The shape of the window shown is a combination of a rectangle and a semicircle. Find an equation of the circle of which
the semicircle is a part.
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 71

Meshing gears For design purposes, the large gear is described by the circle $x^{2}+y^{2}=16 .$ The smaller gear is a circle centered at $(7,0)$ and tangent to the larger circle. Find an equation in standard form of the smaller gear.
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 72

Walkways The walkway shown is bounded by the two circles $x^{2}+y^{2}=2,500$ and $(x-10)^{2}+y^{2}=900,$ measured in feet. Find the largest and the smallest width of the walkway.
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 73

Solar furnaces $A$ parabolic mirror collects rays of the sun and concentrates them at its focus. In the illustration, how far from the vertex of the parabolic mirror will it get the hottest? (All measurements are in feet.)
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 74

Searchlight reflectors A parabolic mirror reflects light in a beam when the light source is placed at its focus. In the illustration, how far from the vertex of the parabolic reflector should the light source be placed? (All measurements are in feet.)
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 75

Writing equations of parabolas Derive an equation of the parabolic arch shown.
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 76

Projectiles The cannonball in the illustration follows the parabolic trajectory $y=30 x-x^{2}$. How far short of the castle docs it land?
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 77

Satellite antennas The cross section of the satellite antenna in the illustration is a parabola given by the equation $y=\frac{1}{16} x^{2},$ with distances measured in feet. If the dish is 8 feet wide, how deep is it?
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 78

Design of a satellite antenna The cross section of the satellite antenna shown is a parabola with the pickup at its focus. Find the distance $d$ from the pickup to the center of the dish.
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 79

Toy rockets A toy rocket is $s$ feet above the Earth at the end of $t$ seconds, where $s=-16 t^{2}+80 \sqrt{3} t$ Find the maximum height of the rocket.

Kirsten Bauck

Numerade Educator

Problem 80

Operating a resort A resort owner plans to build and rent $n$ cabins for $d$ dollars per week. The price $d$ that she can charge for each cabin depends on the number of cabins she builds, where $d=-45\left(\frac{n}{(12)}-\frac{1}{2}\right) .$ Find the number of cabins she should build to maximize her weekly income.

AG

Ankit Gupta

Numerade Educator

Problem 81

Design of a parabolic reflector Find the outer diameter (the length $\overline{A B}$ ) of the parabolic reflector shown.
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 82

Design of a suspension bridge The cable between the towers of the suspension bridge shown in the illustration has the shape of a parabola with vertex 15 feet above the roadway. Find an equation of the parabola.
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 83

Gateway Arch The Gateway Arch in St. Louis has a shape that approximates a parabola. (See the illustration.) Find the width $w$ of the arch 200 feet above the ground.
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 84

Building tunnels A construction firm plans to build a tunnel whose arch is in the shape of a parabola. (Sce the illustration.) The tunnel will span a two lane highway 8 meters wide. To allow safe passage for vehicles, the tunnel must be 5 meters high at a distance of 1 meter from the tunnel's edge. Find the maximum height of the tunnel.
(FIGURE CAN'T COPY)

Kirsten Bauck

Numerade Educator

Problem 85

Show that the standard form of the equation of a parabola $(y-2)^{2}=8(x-1)$ is a special case of the general form of a second-degree equation in two variables.

Kirsten Bauck

Numerade Educator

Problem 86

Show that the standard form of the equation of a circle $(x+2)^{2}+(y-5)^{2}=36$ is a special case of the general form of a second-degree equation in two variables.

Kirsten Bauck

Numerade Educator

Problem 87

Find an equation, in the form $(x-h)^{2}+(y-k)^{2}=r^{2}$ of the circle passing through the given points.
$$(0,8),(5,3), \text { and }(4,6)$$

Kirsten Bauck

Numerade Educator

Problem 88

Find an equation, in the form $(x-h)^{2}+(y-k)^{2}=r^{2}$ of the circle passing through the given points.
$$(-2,0),(2,8), \text { and }(5,-1)$$

Kirsten Bauck

Numerade Educator

Problem 89

Find an equation of the parabola passing through the given points. Gire the equation in the form $y=a x^{2}+b x+c$.
$$(1,8),(-2,-1), \text { and }(2,15)$$

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 90

Find an equation of the parabola passing through the given points. Gire the equation in the form $y=a x^{2}+b x+c$.
$$(1,-3),(-2,12), \text { and }(-1,3)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 91

Ballistics A stone tossed upward is $s$ feet above the Earth after $t$ seconds, where $s=-16 t^{2}+128 t$ Show that the stone's height $x$ seconds after it is thrown is equal to its height $x$ seconds before it hits the ground.

Kirsten Bauck

Numerade Educator

Problem 92

Ballistics Show that the stone in Exercise 91 reaches its greatest height in one-half of the time it takes until it strikes the ground.

Kirsten Bauck

Numerade Educator

Problem 93

Find the number that must be added to make each binomial a perfect square trinomial.
$$x^{2}+4 x+$$____

Kirsten Bauck

Numerade Educator

Problem 94

Find the number that must be added to make each binomial a perfect square trinomial.
$$y^{2}-12 y+$$

Kirsten Bauck

Numerade Educator

Problem 95

Find the number that must be added to make each binomial a perfect square trinomial.
$$x^{2}-7 x+$$__

Kirsten Bauck

Numerade Educator

Problem 96

Find the number that must be added to make each binomial a perfect square trinomial.
$$y^{2}+11 y+$$__

Kirsten Bauck

Numerade Educator

Problem 97

Solve each equation.
$$x^{2}+4 x=5$$

Kirsten Bauck

Numerade Educator

Problem 98

Solve each equation.
$$y^{2}-12 y=13$$

Kirsten Bauck

Numerade Educator

Problem 99

Solve each equation.
$$x^{2}-7 x-18=0$$

Kirsten Bauck

Numerade Educator

Problem 100

Solve each equation.
$$y^{2}+11 y=-18$$

Kirsten Bauck

Numerade Educator