Elementary and Intermediate Algebra

Marvin L. Bittinger, David J. Ellenbogen,Barbara L. Johnson

Chapter 13

Conic Sections

Educators


Problem 1

a) Vertex: $(-2,5)$
b) Vertex: $(5,-2)$
c) Vertex: $(2,-5)$
d) Vertex: $(-5,2)$
e) Center: $(-2,5)$
f) Center: $(2,-5)$
g) Center: $(5,-2)$
h) Center: $(-5,2)$
$$(x-2)^{2}+(y+5)^{2}=9$$

Heather Z.
Numerade Educator

Problem 2

Match the equation with the center or vertex of its graph, listed in the column on the right.
a) Vertex: $(-2,5)$
b) Vertex: $(5,-2)$
c) Vertex: $(2,-5)$
d) Vertex: $(-5,2)$
e) Center: $(-2,5)$
f) Center: $(2,-5)$
g) Center: $(5,-2)$
h) Center: $(-5,2)$
$$(x+2)^{2}+(y-5)^{2}=9$$

Heather Z.
Numerade Educator

Problem 3

Match the equation with the center or vertex of its graph, listed in the column on the right.
a) Vertex: $(-2,5)$
b) Vertex: $(5,-2)$
c) Vertex: $(2,-5)$
d) Vertex: $(-5,2)$
e) Center: $(-2,5)$
f) Center: $(2,-5)$
g) Center: $(5,-2)$
h) Center: $(-5,2)$
$$(x-5)^{2}+(y+2)^{2}=9$$

Heather Z.
Numerade Educator

Problem 4

Match the equation with the center or vertex of its graph, listed in the column on the right.
a) Vertex: $(-2,5)$
b) Vertex: $(5,-2)$
c) Vertex: $(2,-5)$
d) Vertex: $(-5,2)$
e) Center: $(-2,5)$
f) Center: $(2,-5)$
g) Center: $(5,-2)$
h) Center: $(-5,2)$
$$(x+5)^{2}+(y-2)^{2}=9$$

Heather Z.
Numerade Educator

Problem 5

Match the equation with the center or vertex of its graph, listed in the column on the right.
a) Vertex: $(-2,5)$
b) Vertex: $(5,-2)$
c) Vertex: $(2,-5)$
d) Vertex: $(-5,2)$
e) Center: $(-2,5)$
f) Center: $(2,-5)$
g) Center: $(5,-2)$
h) Center: $(-5,2)$
$$y=(x-2)^{2}-5$$

Heather Z.
Numerade Educator

Problem 6

Match the equation with the center or vertex of its graph, listed in the column on the right.
a) Vertex: $(-2,5)$
b) Vertex: $(5,-2)$
c) Vertex: $(2,-5)$
d) Vertex: $(-5,2)$
e) Center: $(-2,5)$
f) Center: $(2,-5)$
g) Center: $(5,-2)$
h) Center: $(-5,2)$
$$y=(x-5)^{2}-2$$

Heather Z.
Numerade Educator

Problem 7

Match the equation with the center or vertex of its graph, listed in the column on the right.
a) Vertex: $(-2,5)$
b) Vertex: $(5,-2)$
c) Vertex: $(2,-5)$
d) Vertex: $(-5,2)$
e) Center: $(-2,5)$
f) Center: $(2,-5)$
g) Center: $(5,-2)$
h) Center: $(-5,2)$
$$x=(y-2)^{2}-5$$

Heather Z.
Numerade Educator

Problem 8

Match the equation with the center or vertex of its graph, listed in the column on the right.
a) Vertex: $(-2,5)$
b) Vertex: $(5,-2)$
c) Vertex: $(2,-5)$
d) Vertex: $(-5,2)$
e) Center: $(-2,5)$
f) Center: $(2,-5)$
g) Center: $(5,-2)$
h) Center: $(-5,2)$
$$x=(y-5)^{2}-2$$

Heather Z.
Numerade Educator

Problem 9

Graph. Be sure to label each vertex.
$$y=-x^{2}$$

Heather Z.
Numerade Educator

Problem 10

Graph. Be sure to label each vertex.
$$y=2 x^{2}$$

Heather Z.
Numerade Educator

Problem 11

Graph. Be sure to label each vertex.
$$y=-x^{2}+4 x-5$$

Heather Z.
Numerade Educator

Problem 12

Graph. Be sure to label each vertex.
$$x=4-3 y-y^{2}$$

Heather Z.
Numerade Educator

Problem 13

Graph. Be sure to label each vertex.
$$x=y^{2}-4 y+2$$

Heather Z.
Numerade Educator

Problem 14

Graph. Be sure to label each vertex.
$$y=x^{2}+2 x+3$$

Heather Z.
Numerade Educator

Problem 15

Graph. Be sure to label each vertex.
$$x=y^{2}+3$$

Heather Z.
Numerade Educator

Problem 16

Graph. Be sure to label each vertex.
$$x=-y^{2}$$

Heather Z.
Numerade Educator

Problem 17

Graph. Be sure to label each vertex.
$$x=2 y^{2}$$

Heather Z.
Numerade Educator

Problem 18

Graph. Be sure to label each vertex.
$$x=y^{2}-1$$

Heather Z.
Numerade Educator

Problem 19

Graph. Be sure to label each vertex.
$$x=-y^{2}-4 y$$

Heather Z.
Numerade Educator

Problem 20

Graph. Be sure to label each vertex.
$$x=y^{2}+3 y$$

Heather Z.
Numerade Educator

Problem 21

Graph. Be sure to label each vertex.
$$y=x^{2}-2 x+1$$

Heather Z.
Numerade Educator

Problem 22

Graph. Be sure to label each vertex.
$$y=x^{2}+2 x+1$$

Heather Z.
Numerade Educator

Problem 23

Graph. Be sure to label each vertex.
$$x=-\frac{1}{2} y^{2}$$

Heather Z.
Numerade Educator

Problem 24

Graph. Be sure to label each vertex.
$$y=-\frac{1}{2} x^{2}$$

Heather Z.
Numerade Educator

Problem 25

Graph. Be sure to label each vertex.
$$x=-y^{2}+2 y-1$$

Heather Z.
Numerade Educator

Problem 26

Graph. Be sure to label each vertex.
$$x=-y^{2}-2 y+3$$

Heather Z.
Numerade Educator

Problem 27

Graph. Be sure to label each vertex.
$$x=-2 y^{2}-4 y+1$$

Heather Z.
Numerade Educator

Problem 28

Graph. Be sure to label each vertex.
$$x=2 y^{2}+4 y-1$$

Heather Z.
Numerade Educator

Problem 29

Find an equation of the circle satisfying the given conditions.
Center $(0,0),$ radius 6

Heather Z.
Numerade Educator

Problem 30

Find an equation of the circle satisfying the given conditions.
Center $(0,0),$ radius 5

Heather Z.
Numerade Educator

Problem 31

Find an equation of the circle satisfying the given conditions.
Center $(7,3),$ radius $\sqrt{5}$

Heather Z.
Numerade Educator

Problem 32

Find an equation of the circle satisfying the given conditions.
Center $(5,6),$ radius $\sqrt{2}$

Heather Z.
Numerade Educator

Problem 33

Find an equation of the circle satisfying the given conditions.
Center $(-4 \text { 3) radius } 4 \text { ) } \sqrt{3}$

Heather Z.
Numerade Educator

Problem 34

Find an equation of the circle satisfying the given conditions.
Center $(-2,7),$ radius $2 \sqrt{5}$

Heather Z.
Numerade Educator

Problem 35

Find an equation of the circle satisfying the given conditions.
Center $(-7,-2),$ radius $5 \sqrt{2}$

Heather Z.
Numerade Educator

Problem 36

Find an equation of the circle satisfying the given conditions.
Center $(-5,-8),$ radius $3 \sqrt{2}$

Heather Z.
Numerade Educator

Problem 37

Find an equation of the circle satisfying the given conditions.
Center $(0,0),$ passing through $(-3,4)$

Heather Z.
Numerade Educator

Problem 38

Center $(0,0),$ passing through $(11,-10)$

Heather Z.
Numerade Educator

Problem 39

Find an equation of the circle satisfying the given conditions.
Center $(-4,1),$ passing through $(-2,5)$

Heather Z.
Numerade Educator

Problem 40

Find an equation of the circle satisfying the given conditions.
Center $(-1,-3),$ passing through $(-4,2)$

Heather Z.
Numerade Educator

Problem 41

Find the center and the radius of each circle. Then graph the circle.
$$x^{2}+y^{2}=64$$

Heather Z.
Numerade Educator

Problem 42

Find the center and the radius of each circle. Then graph the circle.
$$x^{2}+y^{2}=36$$

Heather Z.
Numerade Educator

Problem 43

Find the center and the radius of each circle. Then graph the circle.
$$
(x+1)^{2}+(y+3)^{2}=36
$$

Heather Z.
Numerade Educator

Problem 44

Find the center and the radius of each circle. Then graph the circle.
$$
(x-2)^{2}+(y+3)^{2}=4
$$

Heather Z.
Numerade Educator

Problem 45

Find the center and the radius of each circle. Then graph the circle.
$$
(x-4)^{2}+(y+3)^{2}=10
$$

Heather Z.
Numerade Educator

Problem 46

Find the center and the radius of each circle. Then graph the circle.
$$
(x+5)^{2}+(y-1)^{2}=15
$$

Heather Z.
Numerade Educator

Problem 47

Find the center and the radius of each circle. Then graph the circle.
$$x^{2}+y^{2}=10$$

Heather Z.
Numerade Educator

Problem 48

Find the center and the radius of each circle. Then graph the circle.
$$x^{2}+y^{2}=7$$

Heather Z.
Numerade Educator

Problem 49

Find the center and the radius of each circle. Then graph the circle.
$$(x-5)^{2}+y^{2}=\frac{1}{4}$$

Heather Z.
Numerade Educator

Problem 50

Find the center and the radius of each circle. Then graph the circle.
$$
x^{2}+(y-1)^{2}=\frac{1}{25}
$$

Heather Z.
Numerade Educator

Problem 51

Find the center and the radius of each circle. Then graph the circle.
$$
x^{2}+y^{2}+8 x-6 y-15=0
$$

Heather Z.
Numerade Educator

Problem 52

Find the center and the radius of each circle. Then graph the circle.
$$
x^{2}+y^{2}+6 x-4 y-15=0
$$

Heather Z.
Numerade Educator

Problem 53

Find the center and the radius of each circle. Then graph the circle.
$$
x^{2}+y^{2}-8 x+2 y+13=0
$$

Heather Z.
Numerade Educator

Problem 54

Find the center and the radius of each circle. Then graph the circle.
$$
x^{2}+y^{2}+6 x+4 y+12=0
$$

Heather Z.
Numerade Educator

Problem 55

Find the center and the radius of each circle. Then graph the circle.
$$
x^{2}+y^{2}+10 y-75=0
$$

Heather Z.
Numerade Educator

Problem 56

Find the center and the radius of each circle. Then graph the circle.
$$
x^{2}+y^{2}-8 x-84=0
$$

Heather Z.
Numerade Educator

Problem 57

Find the center and the radius of each circle. Then graph the circle.
$$
x^{2}+y^{2}+7 x-3 y-10=0
$$

Heather Z.
Numerade Educator

Problem 58

Find the center and the radius of each circle. Then graph the circle.
$$
x^{2}+y^{2}-21 x-33 y+17=0
$$

Heather Z.
Numerade Educator

Problem 59

Find the center and the radius of each circle. Then graph the circle.
$$
36 x^{2}+36 y^{2}=1
$$

Heather Z.
Numerade Educator

Problem 60

Find the center and the radius of each circle. Then graph the circle.
$$
4 x^{2}+4 y^{2}=1
$$

Heather Z.
Numerade Educator

Problem 61

Graph using a graphing calculator.
$$x^{2}+y^{2}-16=0$$

Heather Z.
Numerade Educator

Problem 62

Graph using a graphing calculator.
$$4 x^{2}+4 y^{2}=100$$

Heather Z.
Numerade Educator

Problem 63

Graph using a graphing calculator.
$$x^{2}+y^{2}+14 x-16 y+54=0$$

Heather Z.
Numerade Educator

Problem 64

Graph using a graphing calculator.
$$x^{2}+y^{2}-10 x-11=0$$

Heather Z.
Numerade Educator

Problem 65

Does the graph of an equation of a circle include the point that is the center? Why or why not?

Heather Z.
Numerade Educator

Problem 66

Is a point a conic section? Why or why not?

Heather Z.
Numerade Educator

Problem 67

To prepare for Section $13.2,$ review solving quadratic equations (Section $11.1)$
Solve for $x$ or for $y .
$$\frac{y^{2}}{16}=1$$

Heather Z.
Numerade Educator

Problem 68

To prepare for Section $13.2,$ review solving quadratic equations (Section $11.1)$
Solve for $x$ or for $y .
$$\frac{x^{2}}{a^{2}}=1$$

Heather Z.
Numerade Educator

Problem 69

To prepare for Section $13.2,$ review solving quadratic equations (Section $11.1)$
Solve for $x$ or for $y .
$$\frac{(x-1)^{2}}{25}=1$$

Heather Z.
Numerade Educator

Problem 70

To prepare for Section $13.2,$ review solving quadratic equations (Section $11.1)$
Solve for $x$ or for $y .
$$\frac{(y+5)^{2}}{12}=1$$

Heather Z.
Numerade Educator

Problem 71

To prepare for Section $13.2,$ review solving quadratic equations (Section $11.1)$
Solve for $x$ or for $y .
$$\frac{1}{4}+\frac{(y+3)^{2}}{36}=1$$

Heather Z.
Numerade Educator

Problem 72

To prepare for Section $13.2,$ review solving quadratic equations (Section $11.1)$
Solve for $x$ or for $y .
$$\frac{1}{9}+\frac{(x-2)^{2}}{4}=1$$

Heather Z.
Numerade Educator

Problem 73

On a piece of graph paper, draw a line and a point not on the line. Then plot several points that are the same distance from the point and from the line. What shape do the points appear to form? How is this set of points different from a circle?

Heather Z.
Numerade Educator

Problem 74

If an equation has two terms with the same degree, can its graph be a parabola? Why or why not?

Heather Z.
Numerade Educator

Problem 75

Find an equation of a circle satisfying the given conditions.
Center $(3,-5)$ and tangent to (touching at one point) the $y$ -axis

Heather Z.
Numerade Educator

Problem 76

Find an equation of a circle satisfying the given conditions.
Center $(-7,-4)$ and tangent to the $x$ -axis

Heather Z.
Numerade Educator

Problem 77

Find an equation of a circle satisfying the given conditions.
The endpoints of a diameter are $(7,3)$ and $(-1,-3)$

Heather Z.
Numerade Educator

Problem 78

Find an equation of a circle satisfying the given conditions.
Center $(-3,5)$ with a circumference of $8 \pi$ units

Heather Z.
Numerade Educator

Problem 79

Find an equation of a circle satisfying the given conditions.
Find the point on the $y$ -axis that is equidistant from $(2,10)$ and $(6,2)$

Heather Z.
Numerade Educator

Problem 80

Find an equation of a circle satisfying the given conditions.
Find the point on the $x$ -axis that is equidistant from $(-1,3)$ and $(-8,-4)$

Heather Z.
Numerade Educator

Problem 81

The equation $x^{2}+y^{2}=\frac{81}{4},$ where $x$ and $y$ represent the number of meters from the center, can be used to draw the outer circle on a wrestling mat used in International, Olympic, and World Championship wrestling. The equation $x^{2}+y^{2}=16$ can be used to draw the inner edge of the red zone. Find the area of the red zone.
( IMAGE CANNOT COPY)

Heather Z.
Numerade Educator

Problem 82

Snowboarding. Each side edge of the Burton $X 8$ 155 snowboard is an arc of a circle with a "running length" of $1180 \mathrm{mm}$ and a "sidecut depth" of $23 \mathrm{mm}$
(GRAPH CANNOT COPY)
a) Using the coordinates shown, locate the center of the circle. (Hint: Equate distances.)
b) What radius is used for the edge of the board?

Heather Z.
Numerade Educator

Problem 83

The Never Summer Infinity 149 snowboard has a running length of $1160 \mathrm{mm}$ and a sidecut depth of $23.5 \mathrm{mm}$ (see Exercise 82 ). What radius is used for the edge of this snowboard?

Heather Z.
Numerade Educator

Problem 84

The Rossignol Blast ski, when lying flat and viewed from above, has edges that are arcs of a circle. (Actually, each edge is made of two arcs of slightly different radii. The arc for the rear half of the ski edge has a slightly larger radius.)
(IMAGE CANNOT COPY)
a) Using the coordinates shown, locate the center of the circle. (Hint: Equate distances.)
b) What radius is used for the arc passing through
$(0,1.5)$ and $(72,0) ?$

Heather Z.
Numerade Educator

Problem 85

Ace Carpentry needs to cut ean arch for the top of an entranceway. The arch needs to be 8 ft wide and 2 ft high. To draw the arch, the carpenters will use a stretched string with chalk eattached at an end as a compass.
(IMAGE CANNOT COPY)
a) Using a coordinate system, locate the center of the circle.
b) What radius should the carpenters use to draw the arch?

Heather Z.
Numerade Educator

Problem 86

During an archaeological dig, Martina finds the bowl fragment shown below. What was the original diameter of the bowl?
(IMAGE CANNOT COPY)

Heather Z.
Numerade Educator

Problem 87

A ferris wheel has a radius of $24.3 \mathrm{ft} .$ Assuming that the center is $30.6 \mathrm{ft}$ off the ground and that the origin is below the center, as in the following figure, find an equation of the circle.
(IMAGE CANNOT COPY)

Heather Z.
Numerade Educator

Problem 88

Use a graph of the equation $x=y^{2}-y-6$ to approximate to the nearest tenth the solutions of each of the following equations.
a) $y^{2}-y-6=2$ (Hint: Graph $x=2$ on the same set of axes as the graph of $x=y^{2}-y-6 .$
b) $y^{2}-y-6=-3$

Heather Z.
Numerade Educator

Problem 89

Power of a Motor. The horsepower of a certain kind of engine is given by the formula
$$H=\frac{D^{2} N}{2.5}$$
where $N$ is the number of cylinders and $D$ is the diameter, in inches, of each piston. Graph this equation, assuming that $N=6$ (a six-cylinder engine). Let $D$ run from 2.5 to 8

Heather Z.
Numerade Educator

Problem 90

If the equation $x^{2}+y^{2}-6 x+2 y-6=0$ is written as $y^{2}+2 y+\left(x^{2}-6 x-6\right)=0,$ it can be regarded as quadratic in $y .$
a) Use the quadratic formula to solve for $y .$
b) Show that the graph of your answer to part (a) coincides with the graph on p. 944

Heather Z.
Numerade Educator

Problem 91

How could a graphing calculator best be used to help you sketch the graph of an equation of the form $x=a y^{2}+b y+c ?$

Heather Z.
Numerade Educator

Problem 92

Why should a graphing calculator's window be "squared" before graphing a circle?

Heather Z.
Numerade Educator