11 and Circles Introduction to Conics - Tench's Homepage /...

Chapter 11 l Introduction to Conics and Circles 423

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11.1 Conics?Conics as Cross Sections | p. 425

11.2 CirclesWriting Equations of Circles in

General and Standard Form | p. 431

11.3 Your Circle is in my Line, Your Line is in my CircleIntersection of Circles

and Lines | p. 439

11.4 Going Off on a Tangent (Line)Tangent Lines | p. 455

11.5 Circles, Circles, All About CirclesIntersections of Two Circles | p. 465

11.6 Get Into GearCircles and Problem Solving | p. 479

The original Ferris Wheel was designed by George Ferris, Jr., for the Chicago World's Fair in

1893. It stood 80.4 meters (264 feet) tall. Today, the world's tallest Ferris Wheel is the Singapore

Flyer, which stands 165 meters (541 feet) tall. You will use the mathematics of circles to create a

detailed schematic of a Ferris wheel.

Introduction to Conics and Circles11

CHAPTER

11

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424 Chapter 11 l Introduction to Conics and Circles

Lesson 11.1 l Conics as Cross Sections 425

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11Problem 1 When a three-dimensional solid, such as a cube, is cut by a plane, the

two-dimensional figure that results is called a plane section or cross section

of the solid. The shape of the cross section depends on the position of the plane

with respect to the solid.

Four special cross sections called conic sections are formed when a plane

intersects a solid called a double-napped cone. An example of a double-napped

cone is shown. The upper and lower cones are called nappes.

Vertex

Axis

Edge of thecone

Additionally, the intersection of a plane and a double-napped cone may form a point,

a line, or intersecting lines. These cross sections are called degenerate conics.

11.1 Conics?Conics as Cross Sections

ObjectivesIn this lesson you will:

l Define the degenerate conics.

l Define circles, ellipses, hyperbolas, and

parabolas as conic sections.

Key Termsl conic sections

l nappes

l degenerate conics

In this activity, you are going to examine some equations of curves that are among

the oldest aspects of math studied systematically and thoroughly. They are said

to be discovered by Menaechmus (a Greek, c. 375–325 BC), tutor to Alexander the

Great. They were conceived in an attempt to solve the three famous problems of

trisecting the angle, duplicating the cube, and squaring the circle.

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1. Describe how the intersection of a plane and a double-napped cone could

result in a point.


result in a line.


result in intersecting lines.

4. On the figure shown, draw a plane that intersects the double-napped cone

perpendicular to the axis. Then describe the cross section.

Vertex

Axis

Edge of thecone

5. What is the mathematical term for the cross section in Question 4?

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6. Describe the conic section that results when a plane intersects a single nappe

not perpendicular to the axis, but at an angle that is less than the central

angle of the nappe.

Vertex

Axis

Edge of thecone


8. On the figure shown, draw a plane that intersects both nappes of the double-

napped cone parallel to the axis. Then describe the cross section.

Vertex

Axis

Edge of thecone


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10. On the figure shown, draw a plane that intersects one nappe of the double-

napped cone parallel to the edge of the cone. Then describe the cross

section.

Vertex

Axis

Edge of thecone


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When a plane intersects one nappe of a double-napped cone perpendicular to the

axis of the cone, the curve that results is a circle.

Vertex

Axis

Edge of thecone

When a plane intersects a single nappe not perpendicular to the axis, but at

an angle that is less than the central angle of the nappe, the curve that results

is an ellipse.

Vertex

Axis

Edge of thecone

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When a plane parallel to the axis of the cone intersects both nappes of the cone,

the curve that results is a hyperbola.

Vertex

Axis

Edge of thecone

When a plane intersects one nappe of the double-napped cone parallel to the edge

of the cone, the curve that results is a parabola.

Vertex

Axis

Edge of thecone

Apollonius was the first to base the theory of all conics on sections of one circular

cone, right or oblique. He is also the one to give the names ellipse, parabola,

and hyperbola. Since then, the study of conics has been essential. During the

Renaissance, Kepler’s first law of planetary motion, Descartes and Fermat’s

coordinate geometry (analyzing geometric figures using coordinate systems), and

Desargues, La Hire, and Pascal’s projective geometry (in which, for instance, the

projection of a circle onto a plane is a conic section) all relied on the study of

conics. During the last half of the 20th century, conics once more have become

increasingly studied as the basic curves of space travel.

Be prepared to share your methods and solutions.

Lesson 11.2 l Writing Equations of Circles in General and Standard Form 431

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Problem 1 A circle is the set of all points in a plane equidistant from a point called the center.

The distance from a point on the circle to the center is the radius of the circle.

A circle is named by its center. The circle shown is circle A.

A

r

rr

Center

Circle

11.2 CirclesWriting Equations of Circles in General and Standard Form


l Write and graph the equation of a circle

with its center at the origin.

l Write the equation of a circle with its

center at (h, k).

l Transform graphs and equations

of circles.

Key Termsl circle

l locus (loci)

l standard form of the equation of

a circle

l general form of the equation of

a circle

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Previously, you used the distance formula to calculate specific points a given

distance or an equal distance from an axis, a line, or two points.

1. Use the distance formula to determine the equation of all points, (x, y), five

units from the origin.

x

(x, y)

86

2

4

6

8

–2–2

42

5(0, 0)

–4

–4

–6

–6

–8

–8

y

2. Complete the following table for the equation you determined in Question 1.

Then graph this equation to confirm that it represents a circle with center

at the origin and radius 5.

x y

3. Determine the equation of a circle with center at the origin and radius 10.

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at the origin and radius 10.

x y

5. Looking at the graphs in Questions 2 and 4, do the

figures have

a. Line symmetry? If so, for what line(s)?

b. Point symmetry? If so, for which point(s)?

c. Rotational symmetry? If so, for which angles of

rotation?

6. Write the equation of a circle with center at the origin

and radius r.

Take NoteA figure has point symmetry

if the figure can be

rotated 180° about the point

and the resulting figure is

identical to the original figure.

Take NoteA figure has line symmetry

if a line can divide the figure

into two parts that are

reflections of each other in

the line.

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7. Any equation that can be rewritten in this form will

have a graph that is a circle with center at the origin

and radius r. Determine whether each of the following

equations are circles. If the equation is a circle, rewrite

it in the form x2 � y2 � r2, and calculate the radius. If it

is not a circle, explain why not.

a. x2 � 82 � �y2

b. 2x2 � 2y2 � 8 � 0

c. 2x2 � 4y2 � 20 � 0

d. 5x2 � 5y2 � 5 � 0

Take NoteA figure has rotational

symmetry if a

rotation clockwise or

counterclockwise about the

figure's center produced an

image that is identical to the

original figure.

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Problem 2 In coordinate geometry, a collection of points that share a property is called a

locus (plural loci). So, a circle can be defined as the locus of points in the same

plane a given distance from a given point. You will now use the distance formula to

determine the set of all points a given distance from a point other than the origin.

1. Use the distance formula to determine the equation of all points five units

from the point (1, 2).

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

(x, y)5



at (1, 2) and radius 5. (Hint: Use your knowledge of congruence of circles,

symmetry, and transformations to calculate points in the table. Note that

3–4–5 is a Pythagorean triple.)

x y

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3. Use the distance formula to determine the equation of all points four units

from the point (�3, 5).



at (�3, 5) and radius 4. (Hint: Use your knowledge of congruence of circles,

symmetry, and transformations to calculate points in the table.)

x y

The standard form of the equation of a circle is (x � h)2 � (y � k)2 � r2 where r is

the radius and (h, k) is the center. The general form of the equation of a circle is

Ax2 � By2 � Cx � Dy � E � 0 where A � B.

5. Which form is more useful or provides more information about the graph of

the circle?

Transforming an equation from general form to standard form requires completing

the squares and factoring. For instance:

x2 � y2 � 4x � 6y � 12 � 0

x2 � 4x � y2 � 6y � 12

x2 � 4x � 4 � y2 � 6y � 9 � 12 � 4 � 9

(x � 2)2 � ( y � 3)2 � 25

Center: (�2, 3) Radius: 5

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6. Transform the following equations from general form to standard form.

Then state the center and radius.

a. x2 � y2 � 2x � 4y � 4 � 0

b. x2 � y2 � x � 10y � 25 � 0

c. 2x2 � 2y2 � 5x � 8y � 10 � 0

d. x2 � y2 � 10x � 12y � 51 � 0

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7. Is a circle a function? Explain.

8. How is the graph of x2 � y2 � r 2 related to the graph of (x � h)2 � ( y � k)2 � r 2?

9. Complete the following table

Circle with center at origin and radius r

Circle with center at (h, k) and radius r

y

x

r

(x, y)

(0, 0)

y

x

r

(x, y)

(h, k)

Center: Center:

Radius: Radius:

Equation: Equation:


Lesson 11.3 l Intersection of Circles and Lines 439

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Problem 1 The Intersection of a Line and a Circle

1. Consider a system of two linear equations. Describe the possible solutions to

the system. Include a sketch of the lines for each possible solution.

11.3 Your Circle is in my Line, Your Line is in my CircleIntersection of Circles and Lines


l List the possible number of solutions when solving a system of

equations for a line and a circle.

l Solve systems of equations involving a line and a circle algebraically,

graphically, and using technology.

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2. Describe the methods that you learned in order to solve a system of

linear equations.

3. Consider a system of a circle and a line. Describe the possible solutions to

the system. Include a sketch of the circle and line for each possible solution.

4. What is the geometric term for a line that intersects a circle at exactly one point?

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5. What is the geometric term for the point of intersection of a tangent line

and a circle?

6. List as many properties as you can about tangent lines and tangent segments.

Problem 2 Solving Systems Graphically 1. Describe how to sketch a circle using the equation of the circle.

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2. For each system, sketch the circle and line. Then estimate the solution to the

system of equations. Check each solution using the original equations.

a. x2 � y2 � 25

y � 2x � 5

b. x2 � 4x � y2 � 2y � 11

x � 2y � 12

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c. x2 � 4x � y2 � 2y � 12 � 32

4x � 3y � 36

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d. ( x � 4)2 � ( y � 5)2 � 40

y � 3x � 1

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3. What are the advantages and disadvantages of determining a solution using

a sketch?

Problem 3 Solving a System Algebraically A system of a linear equation and a circle can be solved algebraically using

substitution. In order to do this, solve the linear equation for a variable, substitute

the linear equation into the equation of the circle, and solve the resulting equation.

Substitute the variable value into either equation to determine the value of the other

variable. Be sure to check your solution.

Solve each system of equations algebraically. Check each solution using the

original equations.

1. x2 � y2 � 20

y � 2x � 10

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2. x2 � y2 � 10

2x � y � 5

3. 4x2 � 4y2 � 464

y � x � 6

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4. x2 � 6x � y2 � 10y � 12 � 14

y � �x � 4

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5. ( x � 4)2 � ( y � 5)2 � 40

y � 3x � 1

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6. What are the advantages and disadvantages of determining a solution

algebraically?

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Problem 4 Solving a System Using Technology

A system of a linear equation and a circle can also be solved using a graphing

calculator.

To graph a function using a graphing calculator, the function must be entered in the

form Y�. To graph a circle, solve the equation for the variable y. For example, solve

the equation of the circle x2 � y2 � 13 for y.

y2 � 13 � x2

√__

y2 � � √_______

13 � x2

y � � √_______

13 � x2

Notice that when taking the square root of both sides, the result is two different

equations. The graph of a circle is represented by two graphs, a top semicircle and

a bottom semicircle.

Many graphing calculators do not have a square display screen so the graph of a

circle may not look like a circle on the screen when the dimensions of the x- and

y-axes are equal. For example, the circle x2 � y2 � 13 is shown with both the x- and

y-dimensions set from �10 to 10.

The ZSquare, or Zoom Square, function adjusts the dimensions of the x- and

y-axes so that the circle will be displayed as a circle. The circle x2 � y2 � 13 is

shown after using ZSquare. Notice that the x-dimensions are now �15.16129

to 15.16129.

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To solve a system of a linear equation and a circle, graph the circle and the line.

Then determine the coordinates of all points of intersection. To determine a point of

intersection, perform the following steps.

l Press the 2nd button and the TRACE button to access the CALC menu.

l Select 5: Intersect.

l The bottom of the screen will have the display “First curve?” Move the cursor

as close as possible to the intersection point along the first curve and press

ENTER. To switch between graphs, use the up and down arrow keys.

l The bottom of the screen will have the display “Second curve?” Move the

cursor as close as possible to the intersection point along the second

curve and press ENTER. To switch between graphs, use the up and down

arrow keys.

l The bottom of the screen will have the display “Guess?” Move the cursor

as close as possible to the intersection point and press ENTER.

l The x- and y-coordinates of the intersection point will be displayed at the

bottom of the screen. Repeat for any additional intersection points.

1. Solve each system using a graphing calculator.

a. ( x � 4)2 � ( y � 5)2 � 40

y � 3x � 1

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b. x2 � 62x � y2 � 37y � 99

y � 2x � 50

c. 3x2 � 24x � 3y2 � 18y � 70

y � 1 __

2 x � 9.5

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d. ( x � 10)2 � ( y � 10)2 � 169

5x � 12y � �1

2. What are the advantages and disadvantages of determining a solution using a

graphing calculator?

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3. Consider the three methods for solving a system of equations involving a

line and a circle: graphing, using algebra, and using a graphing calculator.

Which method do you prefer? Explain your reasoning.


Lesson 11.4 l Tangent Lines 455

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11.4 Going Off on a Tangent (Line) Tangent Lines


l Write the equation of a line tangent to a circle when given the center of the circle

and the point of tangency.

l Write the equation of a line tangent to a circle when given the equation of the

circle and the point of tangency.

l Write the equation of a circle tangent to a line when given the equation of the line

and the center of the circle.

Problem 1 Write the Equation of a Tangent Line when Given the Center of the Circle and the Point of Tangency

1. The graph of a circle with center (�3, 2) is shown. Sketch a line that is tangent

to the circle and is not horizontal or vertical. Label the line as “Estimated

Tangent Line.” What is the point of tangency?

x12

4

8

12

16

20 –4–4

84–8

(–3, 2)

–8

–12

–12

–16

–16

y

2. Calculate the equation of the tangent line you drew.

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3. Draw a radius from the center of the circle to the point of tangency. What

is the relationship between the slope of the radius drawn to the point of

tangency and the slope of the tangent line?

4. Calculate the slope of the tangent line using the coordinates of the center and

point of tangency.

5. Determine the equation of the tangent line using the slope of the tangent line

and the point of tangency.

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6. Graph the equation of the tangent line you determined in Question 5.

Label the line as “Actual Tangent Line.”

7. How does the equation of the estimated tangent line compare to the equation

of the actual tangent line? Why are the equations different?

8. The graph of a circle with center (2, 1) is shown. Determine the equation of

the line tangent to the circle at the point (5, 2).

x16 20 2412

4

8

12

16

20

24

–4 84–8–4

–8

y

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9. The graph of a circle with center (4, 0.5) is shown. Determine the equation of

the line tangent to the circle at the point (2.5, –3).

x8 10 126

2

4

6

8

10

12

–2 42–4–2

–4

y

10. Explain how to determine the equation of a line tangent to a circle through a

given point.

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Problem 2 Write the Equation of a Tangent Line when Given the Equation of the Circle and the Point of Tangency

It is also possible to determine the equation of a tangent line, if you know the

equation of the circle and the point of tangency.

1. Determine the equation of each tangent line.

a. Circle: ( x � 5)2 � ( y � 8)2 � 193 Point of Tangency: (18, �12.9)

b. Circle: 5x2 � 5y2 � 20x � 10y � 100 Point of Tangency: (�2, 6)

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c. Circle: 3x2 � 6x � 3y2 � 48 Point of Tangency: (0, 4)

d. Circle: x2 � y2 � 4 Point of Tangency: (1, √__

3 )

2. In Problem 1, you determined the equation of a tangent line using the center

of a circle and the point of tangency. In Problem 2, you determined the

equation of a tangent line using the equation of a circle and the point of

tangency. How are these calculations similar?

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Problem 3 Write the Equation of a Circle when Given the Tangent Line and the Center of the Circle

1. Draw a circle tangent to the given line with center (�1, 0). Label the circle as

“Estimated Tangent Circle.”

x86

2

4

6

8

–2–2

42–4

–4

–6

–6

–8

–8

y

2. Can you draw more than one circle that is tangent to the given line and has

the given center? Explain.

3. How can you calculate the point of tangency?

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4. The shortest distance between a point and a line is a perpendicular line segment.

Why is this statement important to determine the equation of the tangent circle?

5. What is the equation for the circle that is tangent to the line segment?

Graph the equation of the circle. Label the circle as “Actual Tangent Circle.”

6. Determine the equation of a circle with the center at (9, 4) and tangent to the

line y � 3 __ 2 x � 16.

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7. Determine the equation of a circle with the center at the origin and tangent to

y � � 1 __ 4 x � 6.


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Lesson 11.5 l Intersections of Two Circles 465

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Problem 1 Intersections of Two Circles 1. Consider a system of two circles. Describe the possible solutions to the

system. Include a sketch for each possible solution.

11.5 Circles, Circles, All About Circles Intersections of Two Circles


l List the possible number of solutions to a system of equations of two circles.

l Solve systems of equations involving two circles algebraically and using a

graphing calculator.

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2. Consider the system of linear equations.

3x � 4y � 10

3x � 2y � �14

Is it more efficient to solve the system using substitution or the elimination

method? Explain.

3. Solve the system of linear equations.

3x � 4y � 10

3x � 2y � �14

4. Consider the system of two circles.

(x � 4)2 � (y � 3)2 � 25

(x � 12)2 � (y � 7)2 � 25

Graph each circle on the coordinate grid.

x8 10 12 146

2

4

6

8

10

12

14

16–2 42–2

–4

y

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5. Estimate the solution to the system of circles using the graph.

6. Solve the system of circles algebraically using the elimination process by

completing the following steps.

a. Rewrite each equation in general form.

b. Eliminate the x2 and y2 terms.

c. Solve the resulting equation for y.

d. Graph this linear equation on the grid from Question 4. How is the graph of

this linear equation related to the graphs of the circles?

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e. Substitute the equation of the line from part (c) into the equation for either

circle. Simplify so that one side of the resulting equation is equal to zero.

f. Solve the equation from part (e) for x.

g. Determine the corresponding y-coordinates for each x-value. Write each

solution as a coordinate pair.

7. Describe the steps to solve a system of two circles using a

graphing calculator.

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8. Solve the following system of circles using a graphing calculator.

(x � 3)2 � (y � 5)2 � 42

5x2 � 15x � 5y2 � 10y � 50

Include a sketch of the graph of the two circles.

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Problem 2 Solving Systems of Equations of Two Circles

Solve each system of two circles algebraically. Then verify the solution(s) using a

graphing calculator. Include a sketch of each system.

1. x2 � y2 � 9

x2 � y2 � 16

Lesson 11.6 l Circles and Problem Solving 479

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Problem 1 Ferris Wheel Bridget has been hired by an amusement park to design a new Ferris wheel.

Before construction can begin, she must submit a detailed schematic of the ride.

Spoke A

x

y

Spoke B

Spoke C

Spoke D

1. Bridget uses a coordinate plane to define the location of the Ferris wheel and

its seats. She places the Ferris wheel at the origin with a radius of 15 cm.

Write an equation for the outer wheel.

2. Spoke A is represented by the equation y � 55 ___ 24

x. Explain why Spoke C is

perpendicular to Spoke A.

11.6 Get Into Gear Circles and Problem Solving

Objective In this lesson you will:

l Solve real-world problems involving circles and tangent lines by modeling

them with diagrams, equations, and graphs.

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3. Write an equation to represent Spoke C.

4. Spoke B is represented by the equation y � � 28 ___ 11

x. Write an equation for

Spoke D.

5. Calculate the coordinates of each point representing a seat on the

Ferris wheel.

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Problem 2 Marching BandThe choreographer of a high school band plans for the marching band to form two

intersecting circles on the football field. Half of the band will form a circle using the

20-yard line as the center. The remaining members of the band will form a circle

using the nearest 40-yard line as the center. Both centers of the circles will be set

at the hash marks at the middle of the football field. Each circle will span 30 yards.

1. Sketch the circles formed by the marching band and the yard lines.

2. The leader of the marching band and captain majorette will be positioned at

the points of intersection. How far a part will they be standing?

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3. The choreographer would like to increase the distance between the

bandleader and majorette so that they are 30 yards apart. She would also like

the centers of the circles to remain the same and the circles to continue to

match in size. What should the radii of the circles be? Explain your process.

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Problem 3 Gear SystemsMichael is drawing a schematic diagram of a gear system. He is beginning with one

gear and a chain that runs alongside the gear as shown.

For better accuracy, he places his sketch on a coordinate plane. The gear is

centered at the origin and has a radius of 5 centimeters. The point of tangency with

the chain is the point (�3, 4).

1. What is the equation of the line representing the chain?

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2. Michael inserts another gear in the gear system. The new gear is centered at

the point (6, 7) and is also tangent with the chain. What is the radius of the

new gear? Explain your process.


11 and Circles Introduction to Conics - Tench's Homepage /...

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Transcript of 11 and Circles Introduction to Conics - Tench's Homepage /...