Intermediate algebra for college students

Robert F Blitzer

Chapter 10

Conic Sections and Systems of Nonlinear Equations - all with Video Answers

Educators

Section 1

Distance and Midpoint Formulas; Circles

Problem 1

In Exercises 1-18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
$(2,3)$ and $(14,8)$

Cory Kuzinski

Cory Kuzinski

Numerade Educator

Problem 2

In Exercises 1-18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
(5, 1) and (8, 5)

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 3

In Exercises 1-18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
$(4,1)$ and $(6,3)$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 4

In Exercises 1-18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
$(2,3)$ and $(3,5)$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 5

In Exercises 1-18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
$(0,0)$ and $(-3,4)$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 6

In Exercises 1-18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
$(0,0)$ and $(3,-4)$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 7

In Exercises 1-18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
(-2,-6) and $(3,-4)$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 8

In Exercises 1-18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
$(-4,-1)$ and $(2,-3)$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 9

In Exercises 1-18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
$(0,-3)$ and $(4,1)$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 10

In Exercises 1-18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
$(0,-2)$ and $(4,3)$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 11

(3.5, 8.2) and (-0.5, 6.2)

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Problem 12

(2.6, 1.3) and (1.6, -5.7)

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Problem 13

$(0,-\sqrt{3})$ and $(\sqrt{5}, 0)$

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Problem 14

$(0,-\sqrt{2})$ and $(\sqrt{7}, 0)$

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Problem 15

$(3 \sqrt{3}, \sqrt{5})$ and $(-\sqrt{3}, 4 \sqrt{5})$

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Problem 16

$(2 \sqrt{3}, \sqrt{6})$ and $(-\sqrt{3}, 5 \sqrt{6})$

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Problem 17

$\left(\frac{7}{3}, \frac{1}{5}\right)$ and $\left(\frac{1}{3}, \frac{6}{5}\right)$

Nick Johnson

Numerade Educator

Problem 18

$\left(-\frac{1}{4},-\frac{1}{7}\right)$ and $\left(\frac{3}{4}, \frac{6}{7}\right)$

Nick Johnson

Numerade Educator

Problem 19

In Exercises 19-30, find the midpoint of the line segment with the given endpoints.
$(6,8)$ and $(2,4)$

Erika Bustos

Numerade Educator

Problem 20

In Exercises 19-30, find the midpoint of the line segment with the given endpoints.
$(10,4)$ and $(2,6)$

Erika Bustos

Numerade Educator

Problem 21

In Exercises 19-30, find the midpoint of the line segment with the given endpoints.
$(-2,-8)$ and $(-6,-2)$

Erika Bustos

Numerade Educator

Problem 22

In Exercises 19-30, find the midpoint of the line segment with the given endpoints.
$(-4,-7)$ and $(-1,-3)$

Erika Bustos

Numerade Educator

Problem 23

(-3,-4) and $(6,-8)$

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Problem 24

(-2, -1) and (-8, 6)

Nick Johnson

Numerade Educator

Problem 25

$\left(-\frac{7}{2}, \frac{3}{2}\right)$ and $\left(-\frac{5}{2},-\frac{11}{2}\right)$

Nick Johnson

Numerade Educator

Problem 26

$\left(-\frac{2}{5}, \frac{7}{15}\right)$ and $\left(-\frac{2}{5},-\frac{4}{15}\right)$

Nick Johnson

Numerade Educator

Problem 27

$(8,3 \sqrt{5})$ and $(-6,7 \sqrt{5})$

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Problem 28

$(7 \sqrt{3},-6)$ and $(3 \sqrt{3},-2)$

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Problem 29

$(\sqrt{18},-4)$ and $(\sqrt{2}, 4)$

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Problem 30

$(\sqrt{50},-6)$ and $(\sqrt{2}, 6)$

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Problem 31

In Exercises 31-40, write the standard form of the equation of the circle with the given center and radius.
Center $(0,0), r=7$

Erika Bustos

Numerade Educator

Problem 32

In Exercises 31-40, write the standard form of the equation of the circle with the given center and radius.
Center $(0,0), r=8$

Vysakh M

Numerade Educator

Problem 33

In Exercises 31-40, write the standard form of the equation of the circle with the given center and radius.
Center $(3,2), r=5$

Vysakh M

Numerade Educator

Problem 34

Center $(2,-1), r=4$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 35

Center $(-1,4), r=2$

Vysakh M

Numerade Educator

Problem 36

Center $(-3,5), r=3$

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Problem 37

Center $(-3,-1), r=\sqrt{3}$

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Problem 38

Center $(-5,-3), r=\sqrt{5}$

Erika Bustos

Numerade Educator

Problem 39

Center $(-4,0), r=10$

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Problem 40

Center $(-2,0), r=6$

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Problem 41

In Exercises 41-48, give the center and radius of the circle described by the equation and graph each equation.
$x^2+y^2=16$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 42

In Exercises 41-48, give the center and radius of the circle described by the equation and graph each equation.
$x^2+y^2=49$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 43

In Exercises 41-48, give the center and radius of the circle described by the equation and graph each equation.
$(x-3)^2+(y-1)^2=36$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 44

In Exercises 41-48, give the center and radius of the circle described by the equation and graph each equation.
$(x-2)^2+(y-3)^2=16$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 45

In Exercises 41-48, give the center and radius of the circle described by the equation and graph each equation.
$(x+3)^2+(y-2)^2=4$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 46

$(x+1)^2+(y-4)^2=25$

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Problem 47

$(x+2)^2+(y+2)^2=4$

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Problem 48

$(x+4)^2+(y+5)^2=36$

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Problem 49

In Exercises 49-56, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.
$x^2+y^2+6 x+2 y+6=0$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 50

In Exercises 49-56, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.
$x^2+y^2+8 x+4 y+16=0$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 51

$x^2+y^2-10 x-6 y-30=0$

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Problem 52

$x^2+y^2-4 x-12 y-9=0$

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Problem 53

$x^2+y^2+8 x-2 y-8=0$

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Problem 54

$x^2+y^2+12 x-6 y-4=0$

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Problem 55

$x^2-2 x+y^2-15=0$

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Problem 56

$x^2+y^2-6 y-7=0$

Sanchit Jain

Numerade Educator

Problem 57

find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding all points of intersection. Check all solutions in both equations.
$\left\{\begin{aligned} x^2+y^2 & =16 \\ x-y & =4\end{aligned}\right.$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 58

find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding all points of intersection. Check all solutions in both equations.
$\left\{\begin{aligned} x^2+y^2 & =9 \\ x-y & =3\end{aligned}\right.$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 59

find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding all points of intersection. Check all solutions in both equations.
$\left\{\begin{aligned}(x-2)^2+(y+3)^2 & =4 \\ y & =x-3\end{aligned}\right.$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 60

find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding all points of intersection. Check all solutions in both equations.
$\left\{\begin{aligned}(x-3)^2+(y+1)^2 & =9 \\ y & =x-1\end{aligned}\right.$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 61

write the standard form of the equation of the circle with the given graph.
Graph can't copy

Erika Bustos

Numerade Educator

Problem 62

write the standard form of the equation of the circle with the given graph.
Graph can't copy

Erika Bustos

Numerade Educator

Problem 63

write the standard form of the equation of the circle with the given graph.
Graph can't copy

Erika Bustos

Numerade Educator

Problem 64

write the standard form of the equation of the circle with the given graph.
Graph can't copy

Erika Bustos

Numerade Educator

Problem 65

a line segment through the center of each circle intersects the circle at the points shown.
a. Find the coordinates of the circle's center.
b. Find the radius of the circle.
c. Use your answers from parts (a) and (b) to write the standard form of the circle's equation.
Graph can't copy

Cory Kuzinski

Numerade Educator

Problem 66

a line segment through the center of each circle intersects the circle at the points shown.
a. Find the coordinates of the circle's center.
b. Find the radius of the circle.
c. Use your answers from parts (a) and (b) to write the standard form of the circle's equation.
Graph can't copy

Cory Kuzinski

Numerade Educator

Problem 67

use the information at the bottom of the previous column to find the distance, to the nearest mile, between each pair of cities.
Boston and San Francisco

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 68

use the information at the bottom of the previous column to find the distance, to the nearest mile, between each pair of cities.
New Orleans and Houston

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 69

A rectangular coordinate system with coordinates in miles is placed with the origin at the eenter of Los Angeles. The figure indicates that the University of Southern California is located 2.4 miles west and 2.7 miles south of central Los Angeles. A seismograph on the campus shows that a small earthquake occurred. The quake's epicenter is estimated to be approximately 30 miles from the university. Write the standard form of the equation for the set of points that could be the epicenter of the quake.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 70

The Ferris wheel in the figure has a radius of 68 feet. The clearance between the wheel and the ground is 14 feet. The rectangular coordinate system shown has its origin on the ground directly below the center of the wheel. Use the coordinate system to write the equation of the circular wheel.

Linh Vu

Numerade Educator

Problem 71

In your own words, describe how to find the distance between two points in the rectangular coordinate system.

Linh Vu

Numerade Educator

Problem 72

In your own words, describe how to find the midpoint of a line segment if its endpoints are known.

Linh Vu

Numerade Educator

Problem 73

What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.

Linh Vu

Numerade Educator

Problem 74

Give an example of a circle's equation in standard form. Describe how to find the center and radius for this circle.

Linh Vu

Numerade Educator

Problem 75

How is the standard form of a circle's equation obtained from its general form?

Casey Castelli

Numerade Educator

Problem 76

Does $(x-3)^2+(y-5)^2=0$ represent the equation of a circle? If not, describe the graph of this equation.

Linh Vu

Numerade Educator

Problem 77

Does $(x-3)^2+(y-5)^2=-25$ represent the equation of a circle? What sort of set is the graph of this equation?

Linh Vu

Numerade Educator

Problem 78

use a graphing utility to graph each circle whose equation is given.
$x^2+y^2=25$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 79

use a graphing utility to graph each circle whose equation is given.
$(y+1)^2=36-(x-3)^2$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 80

use a graphing utility to graph each circle whose equation is given.
$x^2+10 x+y^2-4 y-20=0$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 81

I've noticed that in mathematics, one topic often leads logically to a new topic:

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 82

To avoid sign errors when finding $h$ and $k, ~ I ~ p l a c e$ parentheses around the numbers that follow the subtraction signs in a circle's equation.

Erika Bustos

Numerade Educator

Problem 83

I used the equation $(x+1)^2+(y-5)^2=-4$ to identify the circle's center and radius.

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 84

My graph of $(x-2)^2+(y+1)^2=16$ is my graph of $x^2+y^2=16$ translated two units right and one unit down.

Erika Bustos

Numerade Educator

Problem 85

The equation of the circle whose center is at the origin with radius 16 is $x^2+y^2=16$.

Linh Vu

Numerade Educator

Problem 86

The graph of $(x-3)^2+(y+5)^2=36$ is a circle with radius 6 centered at $(-3,5)$.

Cory Kuzinski

Numerade Educator

Problem 87

The graph of $(x-4)+(y+6)=25$ is a circle with radius 5 centered at $(4,-6)$.

Nick Johnson

Numerade Educator

Problem 88

The graph of $(x-3)^2+(y+5)^2=-36$ is a circle with radius 6 centered at $(3,-5)$.

James Kiss

Numerade Educator

Problem 89

Show that the points $A(1,1+d), B(3,3+d)$, and $C(6,6+d)$ are collinear (lie along a straight line) by showing that the distance from $A$ to $B$ plus the distance from $B$ to $C$ equals the distance from $A$ to $C$.

Linh Vu

Numerade Educator

Problem 90

Prove the midpoint formula by using the following procedure.
a. Show that the distance between $\left(x_1, y_1\right)$ and $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ is equal to the distance between $\left(x_2, y_2\right)$ and $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.
b. Use the procedure from Exercise 89 and the distances from part (a) to show that the points $\left(x_1, y_1\right)$, $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$, and $\left(x_2, y_2\right)$ are collinear.

KS

Kristan Siegel

Numerade Educator

Problem 91

Find all points with $y$-coordinate 2 so that the distance between $(x, 2)$ and $(2,-1)$ is 5 .

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 92

Find the area of the doughnut-shaped region bounded by the graphs of $(x-2)^2+(y+3)^2=25$ and $(x-2)^2+(y+3)^2=36$.

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 93

A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write the point-slope form of the equation of a line tangent to the circle whose equation is $x^2+y^2=25$ at the point $(3,-4)$.

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 94

If $f(x)=x^2-2$ and $g(x)=3 x+4$, find $f(g(x))$ and $g(f(x))$.

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 95

Solve: $2 x=\sqrt{7 x-3}+3$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 96

Solve: $|2 x-5|<10$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 97

Set $y=0$ and find the $x$-intercepts: $\frac{x^2}{9}+\frac{y^2}{4}=1$.

Erika Bustos

Numerade Educator

Problem 98

Set $x=0$ and find the $y$-intercepts: $\frac{x^2}{9}+\frac{y^2}{4}=1$.

Erika Bustos

Numerade Educator

Problem 99

Divide both sides of $25 x^2+16 y^2=400$ by 400 and simplify.

Heather Zimmers

Heather Zimmers

Numerade Educator