11 and circles introduction to conics - tench's homepage /...
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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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426 Chapter 11 l Introduction to Conics and Circles
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1. Describe how the intersection of a plane and a double-napped cone could
result in a point.
2. Describe how the intersection of a plane and a double-napped cone could
result in a line.
3. Describe how the intersection of a plane and a double-napped cone could
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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Lesson 11.1 l Conics as Cross Sections 427
11
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
7. What is the mathematical term for the cross section in Question 6?
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
9. What is the mathematical term for the cross section in Question 8?
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428 Chapter 11 l Introduction to Conics and Circles
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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
11. What is the mathematical term for the cross section in Question 10?
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Lesson 11.1 l Conics as Cross Sections 429
11
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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430 Chapter 11 l Introduction to Conics and Circles
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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
ObjectivesIn this lesson you will:
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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432 Chapter 11 l Introduction to Conics and Circles
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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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Lesson 11.2 l Writing Equations of Circles in General and Standard Form 433
11
4. Complete the following table for the equation you determined in Question 3.
Then graph this equation to confirm that it represents a circle with center
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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434 Chapter 11 l Introduction to Conics and Circles
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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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Lesson 11.2 l Writing Equations of Circles in General and Standard Form 435
11
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
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 (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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436 Chapter 11 l Introduction to Conics and Circles
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3. Use the distance formula to determine the equation of all points four units
from the point (�3, 5).
4. Complete the following table for the equation you determined in Question 3.
Then graph this equation to confirm that it represents a circle with center
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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Lesson 11.2 l Writing Equations of Circles in General and Standard Form 437
11
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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438 Chapter 11 l Introduction to Conics and Circles
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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:
Be prepared to share your methods and solutions.
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
ObjectivesIn this lesson you will:
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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440 Chapter 11 l Introduction to Conics and Circles
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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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Lesson 11.3 l Intersection of Circles and Lines 441
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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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442 Chapter 11 l Introduction to Conics and Circles
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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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Lesson 11.3 l Intersection of Circles and Lines 443
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c. x2 � 4x � y2 � 2y � 12 � 32
4x � 3y � 36
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444 Chapter 11 l Introduction to Conics and Circles
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d. ( x � 4)2 � ( y � 5)2 � 40
y � 3x � 1
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Lesson 11.3 l Intersection of Circles and Lines 445
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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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446 Chapter 11 l Introduction to Conics and Circles
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2. x2 � y2 � 10
2x � y � 5
3. 4x2 � 4y2 � 464
y � x � 6
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Lesson 11.3 l Intersection of Circles and Lines 447
11
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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Lesson 11.3 l Intersection of Circles and Lines 449
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6. What are the advantages and disadvantages of determining a solution
algebraically?
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450 Chapter 11 l Introduction to Conics and Circles
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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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Lesson 11.3 l Intersection of Circles and Lines 451
11
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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Lesson 11.3 l Intersection of Circles and Lines 453
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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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454 Chapter 11 l Introduction to Conics and Circles
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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.
Be prepared to share your methods and solutions.
Lesson 11.4 l Tangent Lines 455
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11.4 Going Off on a Tangent (Line) Tangent Lines
ObjectivesIn this lesson you will:
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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456 Chapter 11 l Introduction to Conics and Circles
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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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Lesson 11.4 l Tangent Lines 457
11
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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458 Chapter 11 l Introduction to Conics and Circles
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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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Lesson 11.4 l Tangent Lines 459
11
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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460 Chapter 11 l Introduction to Conics and Circles
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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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Lesson 11.4 l Tangent Lines 461
11
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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462 Chapter 11 l Introduction to Conics and Circles
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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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Lesson 11.4 l Tangent Lines 463
11
7. Determine the equation of a circle with the center at the origin and tangent to
y � � 1 __ 4 x � 6.
Be prepared to share your methods and solutions.
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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
ObjectivesIn this lesson you will:
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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466 Chapter 11 l Introduction to Conics and Circles
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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
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2. x2 � y2 � 3y � 118
x2 � y2 � 100
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3. x2 � y2 � 4x � 20
x2 � y2 � 64
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4. x2 � y2 � 36
(x � 3)2 � y2 � 9
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5. 2x2 � 20x � 2y2 � 16y � 39
2(x � 5)2 � 2(y � 4)2 � 121
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6. (x � 3)2 � (y � 4)2 � 49
(x � 7)2 � (y � 1)2 � 25
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7. x2 � 16x � y2 � 18y � �45
x2 � 8x � y2 � 6y � �21
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8. x2 � y2 � 16
(x � 5)2 � y2 � 1
Be prepared to share your methods and solutions.
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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.
Be prepared to share your methods and solutions.