Download 4.1 Circles

LESSON
4.1
Name
Circles
Class
4.1
Date
Circles
Essential Question: What is the standard form for the equation of a circle, and what does
the standard form tell you about the circle?
Common Core Math Standards
The student is expected to:
COMMON
CORE
Resource
Locker
A-CED.A.3
Recall that a circle is the set of points in a plane that are a fixed distance, called the radius, from a given point,
called the center.
Mathematical Practices
COMMON
CORE
Deriving the Standard-Form Equation
of a Circle
Explore
Represent constraints by equations or inequalities, ... and interpret
solutions as viable or nonviable options in a modeling context. Also
A-CED.A.2, G-GPE.A.1, G-GPE.B.4
A
MP.7 Using Structure
Language Objective
Work with a partner to match graphs of circles to their equations in
standard form.
The coordinate plane shows a circle with center C(h, k) and radius r.
P(x, y) is an arbitrary point on the circle but is not directly above
or below or to the left or right of C. A(x, k) is a point with the same
x-coordinate as P and the same y-coordinate as C. Explain why
△CAP is a right triangle.
y
Since point A has the same x-coordinate as point P, segment
P (x, y)
r
C (h, k) A (x, k)
PA is a vertical segment. Since point A has the same
x
y-coordinate as point C, segment CA is a horizontal segment.
ENGAGE
This means that segments PA and CA are perpendicular,
Essential Question: What is the
standard form for the equation of a
circle, and what does the standard form
tell you about the circle?
triangle.
PREVIEW: LESSON
PERFORMANCE TASK
View the online Engage. Discuss the photo and the
generally circular nature of radio-signal reception
strength. Then preview the Lesson Performance Task.
B
© Houghton Mifflin Harcourt Publishing Company
Possible answer: The standard form for the equation
2
2
of a circle is (x - h) + (y - k) = r 2, which tells you
that the center is (h, k) and the radius is r.
which means that ∠CAP is a right angle and △CAP is a right
Identify the lengths of the sides of △CAP. Remember that point P is arbitrary, so you
cannot rely upon the diagram to know whether the x-coordinate of P is greater than
or less than h or whether the y-coordinate of P is greater than or less than k, so you
must use absolute value for the lengths of the legs of △CAP. Also, remember that the
length of the hypotenuse of △CAP is just the radius of the circle.
⎜
The length of segment AP is ⎜
C
The length of segment AC is
x-h
The length of segment CP is
r
⎟.
⎟.
y-k
.
Apply the Pythagorean Theorem to △CAP to obtain an equation of the circle.
(x - h ) + (y - k ) =
2
Module 4
2
2
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ges must
EDIT--Chan
DO NOT Key=NL-A;CA-A
Correction
Lesson 1
159
gh “File info”
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Date
Class
Name
Circles
HARDCOVER
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Resource
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A-CED.A.3
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Explore
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4.1
Quest
Essential
COMMON
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must use
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Theorem
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Pythagorean
2
Apply the
2
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k
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
Turn to Lesson 4.1 in the
hardcover edition.
x
⎜
⎜
⎟
⎟
© Houghto
n Mifflin
Harcour t
Publishin
y
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

) (
(
)
Lesson 1
159
Module 4
159
Lesson 4.1
L1.indd
4_U2M04
SE38589
A2_MNLE
159
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3:27 PM
3/27/14 5:58 PM
Reflect
1.
EXPLORE
Discussion Why isn’t absolute value used in the equation of the circle?
Since squaring removes any negative signs just as absolute value does, there’s no need to
Deriving the Standard-Form
Equation of a Circle
take absolute value before squaring.
2.
Discussion Why does the equation of the circle also apply to the cases in which P has the same
x-coordinate as C or the same y-coordinate as C so that △CAP doesn’t exist?
If P has the same x-coordinate as C, then P’s y-coordinate must be either k + r or k - r.
INTEGRATE TECHNOLOGY
So,(x - h) + (y - k) = (h - h) + ((k ± r) - k) = 02 + (±r) = r 2, and the equation
2
2
2
2
2
Students have the option of completing the Explore
activity either in the book or online.
of the circle is still satisfied. Similarly, if P has the same y-coordinate as C, then P’s
x-coordinate must be either h + r or h - r. So, (x - h) + (y - k) = ((h ± r) - h)
2
2
2
+ (k - k) = (±r) + 0 2 = r 2, and the equation of the circle is still satisfied.
2
2
QUESTIONING STRATEGIES
Writing the Equation of a Circle
Explain 1
Why does point A have coordinates (x, k)?
It is below P(x, y), which has x-coordinate x,
and on the same horizontal line as C(h, k), which
has y-coordinate k.
The standard-form equation of a circle with center
C(h, k) and radius r is (x - h) + (y - k) = r 2. If you solve this
___
2
2
equation for r, you obtain the equation r = √(x -h) + (y - k) , which gives you a means for finding the radius of a
2
2
circle when the center and a point P(x, y) on the circle are known.
Example 1

Write the equation of the circle.
The circle with center C(-3, 2) and radius r = 4
Why is it necessary to use absolute value signs
when representing the length of the legs of the
right triangle? Since P could be any point on the
circle, absolute value signs are used to make sure
the length of each leg is a positive number.
Substitute -3 for h, 2 for k, and 4 for r into the general equation and simplify.
(x - (-3))
2
+ (y - 2) = 4 2
2
(x + 3) 2 + (y - 2) 2 = 16

The circle with center C(-4, -3) and containing the point P(2, 5)
r = CP
=
=
=
© Houghton Mifflin Harcourt Publishing Company
Step 1 Find the radius.
―――――――――――――
2
2
) (
√(
√( ) ( )
2 - (-4) +
―――――――
2
2
)
5 - (-3)
6 + 8
―――――
36 + 64
√
_
= √ 100
= 10
Step 2 Write the equation of the circle.
(x - (-4))
2
+ (y - (-3)) = 10
2
EXPLAIN 1
Writing the Equation of a Circle
AVOID COMMON ERRORS
Some students may forget to square the radius when
writing the equation. Others may take its square root.
Help students to avoid making these errors by having
them write r 2 above the place in the equation where
they need to write the square of the radius.
2
(x + 4) 2 + (y + 3) = 100
Module 4
2
160
Lesson 1
PROFESSIONAL DEVELOPMENT
A2_MNLESE385894_U2M04L1.indd 160
Math Background
The equation of a circle is based on the fact that all of the points on the circle are a
fixed distance from a given point. This distance can be found using the Pythagorean
Theorem. In the derivation, the fixed distance, the radius, is represented by the
hypotenuse of a right triangle that has one vertex at the center of the circle, and one
on the circle itself. Applying the Pythagorean Theorem produces the equation of the
2
2
that taking the square root of each side of this
circle, (x - h) + (y - k) = r 2. Note
――――――
2
2
equation produces the equation √(x - h) + (y - k) = r. This equation shows
that the radius is the distance between the two points.
QUESTIONING STRATEGIES
5/22/14 2:22 PM
Do you need to be given the coordinates of the
center to write an equation of a circle? If not,
what other information could you use to find the
center? Explain. No. If you know the endpoints of a
diameter of the circle, you can find the coordinates
of the center using the Midpoint Formula.
Circles
160
Your Turn
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Discuss with students how the graph of the
Write the equation of the circle.
3.
2
(x - 1) 2 + (y - (-4)) = 2 2
2
(x - 1) 2 + (y + 4) = 4
2
equation (x - h) + (y - k) = r is a transformation
of the graph of x 2 + y 2 = r 2. Have students describe
the transformation and compare the two graphs.
2
The circle with center C(1, -4) and radius r = 2
2
4.
The circle with center C(-2, 5) and containing the point P(-2, -1)
Because points C and P have the same x-coordinate, the radius of the circle is just the
absolute value of the difference of their y-coordinates, so r = |5 - (-1)| = |6| = 6.
(x - (-2))
+ (y - 5) = 6 2
2
(x + 2) + (y - 5) 2 = 36
EXPLAIN 2
2
Explain 2
Rewriting an Equation of a Circle to Graph the Circle
Expanding the standard-form equation (x - h) + (y - k) = r results in a general second-degree
2
equation in two variables having the form x 2 + y + cx + dy + e = 0. In order to graph such
an equation or an even more general equation of the form ax 2 + ay 2 + cx + dy + e = 0. you
must complete the square on both x and y to put the equation in standard form and identify
the circle’s center and radius.
Rewriting an Equation of a
Circle to Graph the Circle
2
QUESTIONING STRATEGIES
Example 2
How do you know what number to add to
make perfect square trinomials when
converting to standard form? Take half of the
coefficient of the x term, and square it. Then do the
same with the coefficient of the y term.

2
2
Graph the circle after writing the equation in standard form.
x 2 + y 2 - 10x + 6y + 30 = 0
Write the equation.
(
x 2 + y 2 - 10x + 6y + 30 = 0
Prepare to complete the
x 2 - 10x +
square on x and y.
Complete both squares.
© Houghton Mifflin Harcourt Publishing Company
Once the equation is in standard form, how
do you find the diameter of the circle? The
diameter is 2 times the square root of the number
that represents r 2.
2
) + (y + 6y + ) = -30 +
2
+
(x 2 - 10x + 25) + (y 2 + 6y + 9) = -30 + 25 + 9
(x - 5) 2 + (y + 3) 2 = 4
Factor and simplify.
_
The center of the circle is C(5, -3), and the radius is r = √4 = 2.
Graph the circle.
y
0
-2
x
2
4
6
-4
-6
Module 4
161
Lesson 1
COLLABORATIVE LEARNING
A2_MNLESE385894_U2M04L1.indd 161
Peer-to-Peer Activity
Have students work in pairs. Instruct each student in each pair to write an
equation of a circle in standard form. Have them graph the circles, keeping their
graphs hidden from their partners. Have them also convert their equations to the
general form ax 2 + by 2 + cx + dy + e = 0 by expanding and combining like
terms. Instruct students to exchange the general forms of their equations, write
each other’s equations in standard form, and draw the graph. Have students
compare their work.
161
Lesson 4.1
3/28/14 12:26 PM
B
4x 2 + 4y 2 + 8x - 16y + 11 = 0
AVOID COMMON ERRORS
4x 2 + 4y 2 + 8x - 16y + 11 = 0
Write the equation.
When adding the numbers to the constant term to
maintain the equality, students may forget to multiply
the number added to complete each square by the
leading coefficient that was factored from the variable
terms. Help them to avoid this error by encouraging
them to circle the leading coefficients that have been
factored out in the process of completing the square.
4(x 2 + 2x) + 4(y 2 - 4y) + 11 = 0
Factor 4 from the x terms
and the y terms.
(
) + 4(y
4 x 2 + 2x +
Prepare to complete
the square on x and y.
(
- 4y +
) = -11 + 4( ) + 4( )
) + 4(y - 4y + 4 ) = -11 + 4( 1 ) + 4( 4 )
4(x + 1 ) + 4(y - 2 ) = 9
(x + 1 ) + (y - 2 ) = _
4 x 2 + 2x + 1
Complete both
squares.
2
2
2
Factor and simplify.
2
2
2
9
4
Divide both sides by 4.
(
)
The center is C -1 , 2
Graph the circle.
_
, and the radius is r =
4
√
_9
= _
2 .
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Have students compare how the
3
4
y
2
completing-the-square process is used to write
quadratic functions in vertex form with how it is used
to write a circle in standard form. Have them describe
both the similarities and the differences.
x
-4
-2
0
2
4
-2
-4
Your Turn
Graph the circle after writing the equation in standard form.
5.
y
2
-6
-4
-2
(x 2 + 4x) + (y 2 + 6y) = -4
2
(x + 4x + 4) + (y 2 + 6y + 9) = -4 + 4 + 9
(x + 2) 2 + (y + 3) = 9
2
6.
―
-8
9x + 9y − 54x − 72y + 209 = 0
6
9x 2 + 9y 2 - 54x - 72y + 209 = 0
9(x 2 - 6x) + 9(y 2 - 8y) = -209
9(x - 6x + 9) + 9(y - 8y + 16) = -209 + 9(9) + 9(16)
(x - 3) 2 + (y - 4)
2
16
=_
9
The center is C(3, 4), and the radius is r =
Module 4
――
16 _
4
= .
√_
9
3
162
y
4
2
2
2
9(x - 3) + 9(y - 4) = 16
-2
-6
2
2
x
2
-4
The center is C(-2, -3), and the radius is r = √9 = 3.
2
0
© Houghton Mifflin Harcourt Publishing Company
x 2 + y + 4x + 6y + 4 = 0
x 2 + y 2 + 4x + 6y + 4 = 0
2
x
-2
0
2
4
6
-2
Lesson 1
DIFFERENTIATE INSTRUCTION
A2_MNLESE385894_U2M04L1.indd 162
Kinesthetic Experience
3/24/14 10:44 PM
Prepare a length of string with a marker attached at one end. Display a coordinate
plane. With a tack, tape, or some other means, attach the non-marker end of the
string to an arbitrary point on the plane and draw a circle. Ask students to explain
how they could use a point on the circle and the center to find the length of the
string. Help them to see that they can construct a right triangle whose hypotenuse
is the length of the string, and apply the Pythagorean Theorem to determine the
length of the string.
Circles
162
EXPLAIN 3
A circle in a coordinate plane divides the plane into two regions: points inside the circle and
2
2
points outside the circle. Points inside the circle satisfy the inequality (x - h) + (y - k) < r 2,
2
2
2
while points outside the circle satisfy the inequality (x - h) + (y - k) > r .
Solving a Real-World Problem
Involving a Circle
Example 3

QUESTIONING STRATEGIES
If you know the equation of a circle, how can
you determine whether a given point lies
inside, outside, or on the circle? You can substitute
the coordinates of the point for x and y in the
equation of the circle, and see whether the value
2
2
of (x - h) + (y - k) is less than, greater than, or
equal to the value of r 2. If it is less than r 2, the point
lies inside the circle; if greater than, the point lies
outside the circle; and if equal to, the point lies on
the circle.
inside the circle satisfy the inequality
(x - h)2 + (y - k)2 < r 2. Students should recognize
that a point inside the circle would lie on a circle
whose radius would be shorter than r, and whose
2
2
equation would be (x - h) + (y - k) = r 1 2, with
2
2
r 1 < r. Thus, r 1 2 < r 2 and (x - h) + (y - k) < r 2.
Write an inequality representing the given situation, and draw a circle to
solve the problem.
The table lists the locations of the homes of five friends along with the locations of their favorite
pizza restaurant and the school they attend. The friends are deciding where to have a pizza
party based on the fact that the restaurant offers free delivery to locations within a 3-mile radius
of the restaurant. At which homes should the friends hold their pizza party to get free delivery?
Place
Location
Alonzo’s home
A(3, 2)
Barbara’s home
B(2, 4)
Constance’s home
C(-2, 3)
Dion’s home
D(0, -1)
Eli’s home
E(1, -4)
Pizza restaurant
(-1, 1)
School
(1, -2)
Write the equation of the circle with center (-1, 1)
and radius 3.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jose Luis
Pelaez/Corbis
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Ask students to explain why points that are
Solving a Real-World Problem Involving a Circle
Explain 3
(x - (-1))
2
+ (y - 1) = 3 , or (x + 1) + (y - 1) = 9
2
2
2
2
C
The inequality (x + 1) + (y - 1) < 9 represents the situation.
Plot the points from the table and graph the circle.
2
2
Restaurant
-4
The points inside the circle satisfy the inequality. So, the friends
should hold their pizza party at either Constance’s home or
Dion’s home to get free delivery.

-2
4
y
B
A
2
x
0 D
-2
2
4
School
-4
E
In order for a student to ride the bus to school, the student must live more than
2 miles from the school. Which of the five friends are eligible to ride the bus?
(
( )) 2
( 1) ( 2) 4
( 1) ( 2)
Write the equation of the circle with center
(
x- 1
x-
)
2
2
+ y - -2
2
2
=
2 .
2
+ y+
=
2
The inequality x -
( 1 , -2 ) and radius
+ y+
2
> 4 represents the situation.
Use the coordinate grid in Part A to graph the circle.
The points outside the circle satisfy the inequality. So, Alonzo, Barbara, and Constance
are eligible to ride the bus.
Module 4
163
Lesson 1
LANGUAGE SUPPORT
A2_MNLESE385894_U2M04L1.indd 163
Communicate Math
Have students work in pairs. Provide each pair of students with different graphs of
circles and, on separate note cards or sheets of paper, the equations for those
graphs. The first student chooses a graph and decides which equation goes with it,
then explains why they are a match. The second student repeats the procedure
using another graph and equation.
163
Lesson 4.1
3/28/14 12:30 PM
Reflect
7.
ELABORATE
For Part B, how do you know that point E isn’t outside the circle?
The coordinates of point E are (1, -4). Substituting 1 for x and -4 for y in
(x - 1) + (y + 2) gives (1 - 1) + (-4 + 2) = 0 2 + (-2) = 4, so the coordinates of
2
2
2
2
2
QUESTIONING STRATEGIES
E satisfy the equation of the circle, which means that E is on the circle and not outside it.
How is the equation of a circle related to the
equation a 2 + b 2 = c 2 from the Pythagorean
Theorem? The radius is c, and the lengths of the
legs of the right triangle that has the radius as its
hypotenuse are a and b.
Your Turn
Write an inequality representing the given situation, and draw a circle to solve the problem.
8.
Sasha delivers newspapers to subscribers that live within a 4-block radius of
her house. Sasha’s house is located at point (0, -1). Points A, B, C, D, and E
represent the houses of some of the subscribers to the newspaper. To which
houses does Sasha deliver newspapers?
2
2
(x - 0) + (y - (-1)) = 4 2
2
-4
x 2 + (y + 1) = 16
2
-2
D
The inequality x + (y + 1) < 16 represents the situation.
A
B
x
0
2
-2 Sasha
-4
2
2
y
C
4
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Discuss with students why, in the equation
E
The points inside the circle satisfy the inequality x 2 + (y + 1) <16.
2
So, Sasha delivers to the houses located at points B, D, and E.
ax2 + by 2 + cx + dy + e = 0, a and b must be equal
for the equation to be that of a circle. Focus their
attention on the steps involved in converting the
equation to the standard form of a circle, and on the
roles of a and b in the conversion.
Elaborate
9.
Describe the process for deriving the equation of a circle given the coordinates of its center and its radius.
First, choose an arbitrary point P on the circle. Next, find a third point A that forms a
right triangle with points C and P. Then, use the coordinates of the three points to find
the lengths of segments CA and PA. (The length of segment CP is the circle’s radius.)
Finally, use the Pythagorean Theorem to write an equation of the circle.
10. What must you do with the equation ax + ay + cx + dy + e = 0 in order to graph it?
2
2
standard-form equation you can then identify the circle’s center and radius, which you can
then use to graph the circle.
11. What do the inequalities (x - h) + (y - k) < r and (x - h) + (y - k) > r represent?
2
2
2
2
2
2
2
The inequality (x - h) + (y - k) < r represents points inside the circle with
2
2
equation (x - h) + (y - k) = r , and the inequality (x - h) + (y - k) > r
2
2
2
2
2
2
represents points outside the circle.
SUMMARIZE THE LESSON
© Houghton Mifflin Harcourt Publishing Company
Complete the square on x and y to write the equation in standard form. From the
How can you write the equation of a
circle? You can use the coordinates of the
center for h and k, and the radius for r, in the
2
2
equation (x - h) + (y - k) = r 2.
12. Essential Question Check-In What information must you know or determine in order to write an
equation of a circle in standard form?
You must know the center of the circle and its radius to write an equation of the circle in
standard form. If only the center and a point on the circle are known, you can determine
the radius from those two points.
Module 4
A2_MNLESE385894_U2M04L1.indd 164
164
Lesson 1
3/27/14 5:55 PM
Circles
164
Evaluate: Homework and Practice
EVALUATE
• Online Homework
• Hints and Help
• Extra Practice
Write the equation of the circle.
1.
The circle with C(4, -11) and radius r = 16
(x - h) 2 + (y - k) 2 = r 2
(x - 4) 2 + (y - (-11)) = 16 2
(x - 4) 2 + (y + 11) 2 = 256
2
ASSIGNMENT GUIDE
Concepts and Skills
2.
Practice
The circle with C(-7, -1) and radius r = 13
(x - h) 2 + (y - k) = r 2
2
Explore
Deriving the Standard-Form
Equation of a Circle
(x - (-7))
2
+ (y - (-1)) = 13 2
2
(x + 7) 2 + (y + 1) 2 = 169
Example 1
Writing the Equation of a Circle
Exercises 1–4, 21,
24, 25
Example 2
Rewriting an Equation of a Circle to
Graph the Circle
Exercises 5–12
Example 3
Solving a Real-World Problem
Involving a Circle
Exercises 13–20,
22–23
3.
The circle with center C(-8, 2) and
containing the point P(-1, 6)
r = CP
2
2
(x - (-8))
© Houghton Mifflin Harcourt Publishing Company
QUESTIONING STRATEGIES
――
―
= √2
2
2
―
+ (y - 2) = ( √65 )
2
(x + 8) 2 + (y - 2) 2 = 65
= √1 + 1
2
(x - h) 2 + (y - k) 2 = r 2
2
2
(x - 5) 2 + (y - 9) = ( √―
2)
(x - 5) 2 + (y - 9) 2 = 2
In Exercises 5–12, graph the circle after writing the equation in
standard form.
5.
x 2 + y 2 - 2x - 8y + 13 = 0
x + y - 2x - 8y + 13 = 0
(x 2 - 2x) + (y 2 - 8y) = -13
2
(x - 2x + 1) + (y 2 - 8y + 16) = -13 + 1 + 16
(x - 1) 2 + (y - 4) 2 = 4
The center of the circle is C(1, 4), and the radius is r = √4 = 2.
2
―
Exercise
A2_MNLESE385894_U2M04L1.indd 165
Lesson 4.1
6
y
2
Module 4
165
―――――――
√(4 - 5) 2 + (8 - 9) 2
―――――
2
2
= √(-1) + (-1)
=
2
(x - h) 2 + (y - k) 2 = r 2
The circle with center C(5, 9) and
containing the point P(4, 8)
r = CP
――――――――
√(-1 - (-8)) + (6 - 2)
―――
= √7 + 4
= √―――
49 + 16
= √―
65
=
How can you find the radius of a circle if you
know the endpoints of a diameter of the
circle? You can use the distance formula to find the
length of the diameter and take half of that
distance.
4.
4
2
x
-2
0
2
4
-2
Lesson 1
165
Depth of Knowledge (D.O.K.)
6
COMMON
CORE
Mathematical Practices
1–12
1 Recall of Information
MP.7 Using Structure
13–20
2 Skills/Concepts
MP.4 Modeling
21
2 Skills/Concepts
MP.7 Using Structure
22–23
3 Strategic Thinking
MP.4 Modeling
24–25
3 Strategic Thinking
MP.2 Reasoning
3/27/14 5:55 PM
Graph the circle after writing the equation in standard form.
6.
AVOID COMMON ERRORS
x 2 + y 2 + 6x - 10y + 25 = 0
y
x 2 + y 2 + 6x - 10y + 25 = 0
(x 2 + 6x) + (y 2 - 10y) = -25
(x 2 + 6x + 9) + (y 2 - 10y + 25) = -25 + 9 + 25
(x + 3) 2 + (y - 5) 2 = 9
The center of the circle is C(-3, 5), and the radius is r = √9 = 3.
―
7.
4
2
x
-6 -4
-2
y
x + y + 4x + 12y + 39 = 0
(x 2 + 4x) + (y 2 + 12y) = -39
2
(x + 4x + 4) + (y 2 + 12y + 36) = -39 + 4 + 36
(x + 2) 2 + (y + 6) 2 = 1
The center of the circle is C(-2, -6), and the radius is r = √1 = 1.
2
-6
-4
-2
x 2 + y 2 - 8x + 4y + 16 = 0
x
0
―
A2_MNLESE385894_U2M04L1.indd 166
166
4
6
2
4
-4
-6
6
y
4
2
x
-2
0
-2
© Houghton Mifflin Harcourt Publishing Company
8x 2 + 8y 2 - 16x - 32y - 88 = 0
x 2 + y 2 - 2x - 4y - 11 = 0
(x 2 - 2x) + (y 2 - 4y) = 11
(x 2 - 2x + 1) + (y 2 - 4y + 4) = 11 + 1 + 4
(x - 1) 2 + (y - 2) 2 = 16
The center of the circle is C(1, 2), and the radius is r = √16 = 4.
2
-2
8x 2 + 8y 2 - 16x - 32y - 88 = 0
Module 4
y
2
―
9.
-2
-6
2
x + y - 8x + 4y + 16 = 0
(x 2 - 8x) + (y 2 + 4y) = -16
2
(x - 8x + 16) + (y 2 + 4y + 4) = -16 + 16 + 4
(x - 4) 2 + (y + 2) 2 = 4
The center of the circle is C(4, -2), and the radius is r = √4 = 2.
2
x
0
-4
―
8.
0
x 2 + y 2 + 4x + 12y + 39 = 0
2
Students may forget to factor out the leading
coefficients of x 2 and y 2 before completing the square.
Reinforce that the coefficient of each squared term
must be 1 when completing each square.
6
Lesson 1
3/27/14 5:55 PM
Circles
166
10. 2x 2 + 2y 2 + 20x + 12y + 50 = 0
VISUAL CUES
2x 2 + 2y 2 + 20x + 12y + 50 = 0
x 2 + y 2 + 10x + 6y + 25 = 0
(x 2 + 10x) + (y 2 + 6y) = -25
2
(x + 10x + 25) + (y 2 + 6y + 9) = -25 + 25 + 9
(x + 5) 2 + (y + 3) 2 = 9
The center of the circle is C(-5, -3), and the radius is r = √9 = 3.
Suggest that students circle the numbers being added
to or subtracted from x and y, and circle the
preceding addition or subtraction signs, to remind
them to take the opposites of these numbers when
identifying the coordinates of the center of the circle.
-8
-6
-4
-2
-6
-8
11. 12x 2 + 12y 2 - 96x - 24y + 201 = 0
y
12x 2 + 12y 2 - 96x - 24y + 201 = 0
12(x 2 - 8x) + 12(y 2 - 2y) = -201
12(x 2 - 8x + 16) + 12(y 2 - 2y + 1) = -201 + 12(16) + 12(1)
12(x - 4) 2 + 12(y - 1) 2 = 3
1
(x - 4) 2 + (y - 1) 2 = _
4
1.
1
The center of the circle is C(4, 1), and the radius is r = _
=_
4
2
―
√
on a graphing calculator. Lead them to
observe that a circle is not a function, so it cannot be
entered on the Y = screen as one rule. Use an
2
2
equation such as (x + 2) + (y - 3) = 4 to show
how the equation can be solved for y and entered as
two functions: the top half of the circle,
-2
-4
―
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Technology
MP.5 Students may ask how to graph a circle
y
0 x
2
1
0
1
2
4x
3
12. 16x 2 + 16y 2 + 64x - 96y + 199 = 0
y
16x 2 + 16y 2 + 64x - 96y + 199 = 0
16(x 2 + 4x) + 16(y 2 - 6y) = -199
2
16(x + 4x + 4) + 16(y 2 - 6y + 9) = -199 + 16(4) + 16(9)
16(x + 2) 2 + 16(y - 3) 2 = 9
9
(x + 2) 2 + (y - 3) 2 = _
16
3
9
The center of the circle is C(-2, 3), and the radius is r = __
= _.
16
4
4 - (x + 2) ), and the bottom half of the
(y = 3 + √―――――
―――――
circle, (y = 3 - √4 - (x + 2) ).
2
―
√
© Houghton Mifflin Harcourt Publishing Company
2
In Exercises 13–20, write an inequality representing the problem, and
draw a circle to solve the problem.
13. A router for a wireless network on a floor of an office building has a
range of 35 feet. The router is located at the point (30, 30). The lettered
points in the coordinate diagram represent computers in the office. Which
computers will be able to connect to the network through the router?
(x - 30) 2 + (y - 30) 2 = 35 2
(x - 30) 2 + (y - 30) 2 = 1225
The inequality (x - 30) 2 + (y - 30) 2 ≤ 1225 represents the situation.
6
4
2
x
-6
80
60
-4
-2
y
0
C
B
G
D
40
20
0
F
Router
A
E
20
40
x
60
The points on or inside the circle satisfy the inequality. So, the computers
located at points A, B, D, E, and F will be able to connect to the network.
Module 4
A2_MNLESE385894_U2M04L1.indd 167
167
Lesson 4.1
167
Lesson 1
3/28/14 12:52 PM
Write an inequality representing the problem, and draw a circle to solve the
problem.
14. The epicenter of an earthquake is located at the point (20, -30). The
earthquake is felt up to 40 miles away. The labeled points in the coordinate
diagram represent towns near the epicenter. In which towns is the
earthquake felt?
CONNECT VOCABULARY
-20
(x - 20) 2 + (y - (-30)) 2 = 40 2
(x - 20) 2 + (y + 30) 2 = 1600
The inequality (x - 20) 2 + (y + 30) 2 ≤ 1600 represents the
situation.
The points inside the circle satisfy the inequality. So, the
earthquake is felt in the towns located at points B, D, and F.
15. Aida’s cat has disappeared somewhere in her apartment. The last time she
saw the cat, it was located at the point (30, 40). Aida knows all of the cat’s
hiding places, which are indicated by the lettered points in the coordinate
diagram. If she searches for the cat no farther than 25 feet from where she
last saw it, which hiding places will she check?
(x - 30) 2 + (y - 40) 2 = 25 2
(x - 30) 2 + (y - 40) 2 = 625
The inequality (x - 30) 2 + (y - 40) 2 ≤ 625 represents the situation.
Have students label the parts for the equation of a
circle in standard form, identifying the parts that
indicate the coordinates of the center and the radius.
y
A
0
C x
60
20 40
B
-20
Epicenter
-40
D
-60
F
E
y
80
E
60
B
40
D
F
A
G
20
x
C
0
20
A
8
40
60
The points on or inside the circle satisfy the inequality. So, Aida will
search for the cat in its hiding places at points A, B, and D.
(x - (-2)) 2 + (y - 2) 2 = 4 2
(x + 2) 2 + (y - 2) 2 = 16
The inequality (x + 2) 2 + (y - 2) 2 ≤ 16 represents the situation.
The points on or inside the circle satisfy the inequality. So, the
music can be heard at the campsites located at points D, E, F and H.
Module 4
A2_MNLESE385894_U2M04L1 168
168
H
C
D
-8
-4
L
4
F
0
y
B
E
G
-4
-8
x
4
8
J
K
© Houghton Mifflin Harcourt Publishing Company • Image Credits
©wonderlandstock/Alamy:
16. A rock concert is held in a large state park. The concert stage is located
at the point (-2, 2), and the music can be heard as far as 4 miles away.
The lettered points in the coordinate diagram represent campsites within
the park. At which campsites can the music be heard?
Lesson 1
10/17/14 5:13 PM
Circles
168
17. Business When Claire started her in-home computer service and support
business, she decided not to accept clients located more than 10 miles
from her home. Claire’s home is located at the point (5, 0), and the lettered
points in the coordinate diagram represent the homes of her prospective
clients. Which prospective clients will Claire not accept?
(x - 5) 2 + (y - 0) 2 = 10 2
(x - 5) 2 + y 2 = 100
2
The inequality (x - 5) + y 2 > 100 represents the situation.
F
20
B
y
C
10
A
-20 -10
0 D 10
-10
G
x
20
E
-20
The points outside the circle satisfy the inequality. So, Claire
should not accept the prospective clients located at points B, C, E, and F.
18. Aviation An airport’s radar system detects
airplanes that are in flight as far as 60 miles from
the airport. The airport is located at (-20, 40).
The lettered points in the coordinate diagram
represent the locations of airplanes currently in
flight. Which airplanes does the airport’s
radar system detect?
(x - (-20))
B
x
A
-80 -40
2
y
G
D
+ (y - 40) = 60 2
(x + 20) 2 + (y - 40) 2 = 3600
2
2
The inequality (x + 20) + (y - 40) ≤ 3600
represents the situation.
2
80
J
0
-40
F
40
K
C
-80
80
E
H
© Houghton Mifflin Harcourt Publishing Company • Image Credits ©Mikael
Damkier/Shutterstock
The points on or inside the circle satisfy the inequality. So, the airport’s
radar system detects the airplanes at points A, D, G, and J.
19. Due to a radiation leak at a nuclear power plant, the towns up to a
distance of 30 miles from the plant are to be evacuated. The nuclear
power plant is located at the point (-10, -10). The lettered points in
the coordinate diagram represent the towns in the area. Which towns
are in the evacuation zone?
+ (y - (-10)) = 30 2
(x + 10) 2 + (y + 10) 2 = 900
2
2
The inequality (x + 10) + (y + 10) ≤ 900 represents the
situation.
The points on or inside the circle satisfy the inequality. So,
the towns located at points B and E are in the evacuation zone.
(x - (-10))
Module 4
A2_MNLESE385894_U2M04L1 169
169
Lesson 4.1
2
2
169
D
40
20
-40 -20 0
B
-20
C
-40
y
A
E
x
20
40
F
Lesson 1
10/17/14 5:14 PM
20. Bats that live in a cave at point (-10, 0) have a feeding range of 40 miles.
The lettered points in the coordinate diagram represent towns near the
cave. In which towns are bats from the cave not likely to be observed?
Write an inequality representing the problem, and draw a circle to solve
the problem.
(x - (-10))
2
+ (y - 0) = 40
2
2
(x + 10) 2 + y 2 = 1600
2
The inequality (x + 10) + y 2 > 1600 represents the situation.
40
C
B
-40 -20
C(9, -11); r = 13
B. x + y - 18x + 22y + 33 = 0
C(9, 11); r = 15
C. 25x 2 + 25y 2 - 450x - 550y - 575 = 0
C(-9, -11); r = 15
2
2
0
-20
D
The points outside the circle satisfy the inequality. So, bats from
the cave are not likely to be observed in the towns located at
points A and D.
21. Match the equations to the center and radius of the circle each represents.
Show your work.
A. x 2 + y 2 + 18x + 22y - 23 = 0
20
y
A
G
20
F
x
40
E
-40
C(-9, 11); r = 13
D. 25x + 25y + 450x - 550y + 825 = 0
x 2 + y 2 + 18x + 22y - 23 = 0
A
(x 2 + 18x) + (y 2 + 22y) = 23
2
2
(x 2 + 18x + 81) + (y 2 + 22y + 121) = 23 + 81 + 121
(x + 9) 2 + (y + 11) 2 = 225
――
The center of the circle is C(-9, -11), and the radius is r = √225 = 15.
x 2 + y 2 - 18x + 22y + 33 = 0
(x 2 - 18x) + (y 2 + 22y) = -33
2
(x - 18x + 81) + (y 2 + 22y + 121) = -33 + 81 + 121
(x - 9) 2 + (y + 11) 2 = 169
The center of the circle is C(9, -11), and the radius is r = √169 = 13.
B
© Houghton Mifflin Harcourt Publishing Company
――
25x 2 + 25y 2 - 450x - 550y - 575 = 0
x 2 + y 2 - 18x - 22y - 23 = 0
(x 2 - 18x) + (y 2 - 22y) = 23
2
(x - 18x + 81) + (y 2 - 22y + 121) = 23 + 81 + 121
(x - 9) 2 + (y - 11) 2 = 225
The center of the circle is C(9, 11), and the radius is r = √225 = 15.
C
――
25x 2 + 25y 2 + 450x - 550y + 825 = 0
x 2 + y 2 + 18x - 22y = -33
2
(x + 18x) + (y 2 - 22y) = -33
2
(x + 18x + 81) + (y 2 - 22y + 121) = -33 + 81 + 121
(x + 9) 2 + (y - 11) 2 = 169
The center of the circle is (-9, 11), and the radius is r = √169 = 13.
D
――
Answers: B, C, A, D
Module 4
A2_MNLESE385894_U2M04L1.indd 170
170
Lesson 1
5/22/14 2:23 PM
Circles
170
H.O.T. Focus on Higher Order Thinking
22. Multi-Step A garden sprinkler waters the plants in a garden
within a 12-foot spray radius. The sprinkler is located at the
point (5, -10). The lettered points in the coordinate diagram
represent the plants. Use the diagram for parts a–c.
a. Write an inequality that represents the region that does
not get water from the sprinkler. Then draw a circle and
use it to identify the plants that do not get water from the
sprinkler.
20
y
E
10
A
G
C
-30 -20 -10
Circle 3
B
0
-10
10
F
D
-20
(x - 5) + (y - (-10)) = 12
(x - 5) 2 + (y + 10) 2 = 144
2
2
The inequality (x - 5) + (y + 10) > 144 represents the situation.
2
2
2
Circle 2
20
x
30
Circle 1
The points outside the circle satisfy the inequality. So, the plants located
at points A, B, C, E, and G do not get water from the sprinkler.
b. Suppose a second sprinkler with the same spray radius is placed at the point
(10, 10). Write a system of inequalities that represents the region that does not get
water from either sprinkler. Then draw a second circle and use it to identify the
plants that do not get water from either sprinkler.
(x - 10) 2 + (y - 10) 2 = 12 2
(x - 10) 2 + (y - 10) 2 = 144
2
2
The system of inequalities (x - 5) + (y + 10) > 144
2
2
and (x - 10) + (y - 10) > 144 represents the situation.
The points outside both circles satisfy the system of inequalities.
So, the plants located at points A, B, and C would not get water
from either sprinkler.
© Houghton Mifflin Harcourt Publishing Company
c.
Locate the sprinkler at the point (-15, 0).
2
(x - (-15)) + (y - 0) 2 = 12 2
(x + 15) 2 + y 2 = 144
2
2
The system of inequalities (x - 5) + (y + 10) > 144 and
(x - 10) 2 + (y - 10) 2 > 144 and (x + 15) 2 + y 2 > 144 represents the
situation. The points outside all three circles satisfy the system of inequalities.
So, there are no plants that would not get watered by any sprinkler.
Module 4
A2_MNLESE385894_U2M04L1.indd 171
171
Lesson 4.1
Where would you place a third sprinkler with the same spray radius so all the
plants get water from a sprinkler? Write a system of inequalities that represents
the region that does not get water from any of the sprinklers. Then draw a third
circle to show that every plant receives water from a sprinkler.
171
Lesson 1
5/22/14 2:23 PM
23. Represent Real-World Situations The orbit of the planet
Venus is nearly circular. An astronomer develops a model for
the orbit in which the Sun has coordinates S(0, 0), the circular
orbit of Venus passes through V(41, 53), and each unit of the
coordinate plane represents 1 million miles. Write an equation
for the orbit of Venus. How far is Venus from the sun?
Since the center of the orbit is the Sun, the radius of the
orbit is SV.
r = SV
= √(41 - 0) 2 + (53 - 0) 2
= √41 2 + 53 2
= √1681 + 2809
= √4490
≈ 67
―――――――
――――
―――――
――
So, the equation of the orbit is x 2 + y 2 = 67 2, or x 2 + y 2 = 4489,
and Venus is approximately 67 million miles from the Sun.
24. Draw Conclusions The unit circle is defined as the circle with radius 1 centered at
the origin. A Pythagorean triple is an ordered triple of three positive integers, (a, b, c),
that satisfy the relationship a 2 + b 2 = c 2. An example of a Pythagorean triple
is (3, 4, 5). In parts a–d, you will draw conclusions about Pythagorean triples.
a. Write the equation of the unit circle.
(x - h) + (y - k) = r 2
2
2
(x - 0)2 + (y - 0)2 = 1 2
© Houghton Mifflin Harcourt Publishing Company• Image Credits: ©Digital
Vision/Getty Images
x2 + y2 = 1
b. Use the Pythagorean triple (3, 4, 5) and the symmetry of a circle to identify the
coordinates of two points on the part of the unit circle that lies in Quadrant I.
Explain your reasoning.
() ()
2
2
32
52
42
Dividing both sides of 32 + 42 = 52 by 52 gives __
+ __
= __
, or __35 + __45 = 1,
52
52
52
( )
( )
so the points __35 , __45 and __45 , __35 are on the unit circle in Quadrant I.
Module 4
A2_MNLESE385894_U2M04L1.indd 172
172
Lesson 1
5/22/14 2:23 PM
Circles
172
c.
PEERTOPEER DISCUSSION
Ask students to discuss with a partner how they can
tell by inspecting a circle in the form
ax 2 + ay 2 + cx + dy + e = 0 whether the center of
the circle lies on either the x- or y-axis. The circle lies
on the x-axis if d = 0. It lies on the y-axis if c = 0. It
lies on both axes (at the origin) if both c = 0
and d = 0.
Use your answer from part b and the symmetry of a circle to identify the
coordinates of six other points on the unit circle. This time, the points
should be in Quadrants II, III, and IV.
Reflecting the points
(__35 , __45 ) and (__45 , __35 ) across the y-axis gives the points
Reflecting the points
(__35 , __45 ) and (__45 , __35 ) across the x-axis gives the points
(-__35 , __45 ) and (-__45 , __35 ).
(
3
__
, - __45
5
) and (__
4
3
, -__
5
5
).
( )
3
4
4
__
__
__
(- 5 , - 5 ) and (- 5 , -__35 ).
Reflecting the points __35 , __54 and
(__45 , __35 ) across both axes gives the points
d. Find a different Pythagorean triple and use it to identify the coordinates of eight
points on the unit circle.
Answers will vary. Sample answer: The Pythagorean triple (5, 12, 13)
5 ___
12 ___
, 5 , - ___
, 12 ,
( ) (___
13 13 ) ( 13 13 )
5
5
12 ___
12
12
12
5
5
12
, 5 , ___
, - ___
, ___
, - ___
, -___
, - ___
, and (- ___
, - ___
.
(-___
13 13 ) ( 13
13 ) ( 13
13 ) ( 13
13 )
13
13 )
JOURNAL
5 __
generates these eight points: __
, 12 ,
13 13
Have students describe how they can determine
whether a point P lies on a circle if they know the
radius of the circle and the coordinates of the center
of the circle.
25. Make a Conjecture In a two-dimensional plane, coordinates are given by ordered
pairs of the form (x, y). You can generalize coordinates to three-dimensional space
by using ordered pairs of the form (x, y, z) where the coordinate z is used to indicate
displacement above or below the xy-plane. Generalize the standard-form equation of
a circle to find the general equation of a sphere. Explain your reasoning.
Let the center of the sphere be C(h, k, j), the radius be r, and an arbitrary
point on the sphere be P(x, y, z). The plane z = j includes the points
C(h, k, j) and (P’ )(x, y, j), which is the perpendicular projection of
P(x, y, z) onto the plane. Because C and P’ are both in the plane z = j,
© Houghton Mifflin Harcourt Publishing Company
CP’ =
2
(CP’) 2 + (P’P) 2 = (CP) 2
(x - h) +( y - k) ) + (z - j) = r
( √―――――――
2
2
2
2
2
(x - h) 2 + (y - k) 2 + (z - j) 2 = r 2
A2_MNLESE385894_U2M04L1.indd 173
Lesson 4.1
2
∆CP’P, which is a right triangle, gives the following:
Module 4
173
―――――――
+ (y - k) . Applying the Pythagorean Theorem to
√(x - h)
173
Lesson 1
11/12/14 9:10 PM
Lesson Performance Task
QUESTIONING STRATEGIES
A highway that runs straight east and west passes 6 miles south of a radio tower.
The broadcast range of the station is 10 miles.
How do you know whether the beginning and
ending points for the car are within
broadcasting range? Each point is 6 units vertically
below the center point and at the end of a radius
10 units from the center. There are only two
such points.
N
a. Determine the distance along the highway that a car will be
within range of the radio station’s signal.
10 miles
b. Given that the car is traveling at a constant speed of 60 miles per hour,
determine the amount of time the car is within range of the signal.
Radio tower
6 miles
a. The radius of the circle representing the broadcasting
range is 10. Let the position of the radio tower be
(0, 0). Then the highway passes through (0, -6) and so is represented
by the line y = -6.
AVOID COMMON ERRORS
Students who don’t write the distance-rate-time
formula may not take care in determining the
amount of time that the car is within range of the
signal, and thus using the reciprocal value, dividing
60 by 16 instead of the reverse. Remind them to
structure their calculations and use formulas instead
of just operating on numbers.
Write the equation of the circle representing the range of the radio
station’s signal.
(x - 0) 2 + (y - 0) 2 = 10 2
x 2 + y 2 = 100
The highway intersects the circle at points where y = -6.
x 2 + (-6) = 100
x 2 + 36 = 100
x 2 = 64
x = ± √64
x = ±8
2
―
So, the highway intersects the circle at (8, -6) and (-8, -6).
© Houghton Mifflin Harcourt Publishing Company
The distance between the intersection points (8, -6) and (-8, -6)
is 8 - (-8) = 16 miles. So, the car will be within range of the radio
station’s signal for 16 miles.
b. d = rt
d
t= _
r
16
= __
60
4
= __
15
4
So, the car is within range of the signal for __
hour, or 16 minutes.
15
Module 4
Lesson 1
174
EXTENSION ACTIVITY
A2_MNLESE385894_U2M04L1.indd 174
Have students consider a second highway that runs parallel to the first, 2 miles
south of (below) the original highway. Ask how fast a car would need to go along
this highway to be in range of the radio signal for the same amount of time as the
first car? The second car would be in range for 2(√10 2 - 8 2 ) = 12 miles. The
12
second car would need to travel
(60) = 45 miles per hour.
_
3/27/14 5:55 PM
―――
16
Students could also research radio signals and ask whether they do have a circular
range, and what conditions affect both AM and FM signals, either decreasing
signal range (terrain, for example) or allowing a signal to be received a long
distance from its source.
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Circles
174