Lesson 11
COMMON CORE MATHEMATICS CURRICULUM
M5
GEOMETRY
Name_____________________________
Date__________________________
Lesson 11: Recognizing Equations of Circles
Classwork
Opening Exercise
a.
Express this as a trinomial: (𝑥 − 5)2 .
𝑥
b.
2
3
3𝑥
𝑥
𝑥
3
3𝑥
?
𝑥
𝑥
𝑥2
3
3𝑥
3
3𝑥
9
Express this as a trinomial: (𝑥 + 4)2 .
c.
Factor the trinomial: 𝑥 2 + 12𝑥 + 36.
d.
Complete the square to solve the following equation: 𝑥 2 + 6𝑥 = 40.
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
=
=
40
49
1
COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
M5
GEOMETRY
This method of "forcing" the existence of a perfect square trinomial is completing the
square.
Steps for Completing the Square:
1.
Be sure that the coefficient of the highest power is
one.
If it is not, divide each term by that value to create a
leading coefficient of one.
2.
Move the constant term to the right hand side.
3.
Prepare to add the needed value to create the
perfect square trinomial. Be sure to balance the
equation. The boxes may help you remember to
balance.
4.
To find the needed value for the perfect square
trinomial, take half of the coefficient of the middle
term(x-term), square it, and add that value to both sides
of the equation.
5.
Factor the perfect square trinomial.
6.
Take the square root of each side and
solve. Remember to consider both plus and minus
results.
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
2
COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
M5
GEOMETRY
Practice this:
Find the value that completes the square and then rewrite the equation as a perfect square. Then solve the equation by
taking the square root of each side and solving for the variable:
1.
𝑥 2 − 6𝑥 + _______
2.
𝑥 2 + 36𝑥 + _______
3.
𝑥 2 − 20𝑥 + _______
4.
𝑥 2 + 28𝑥 + _______
5.
𝑥 2 − 28𝑥 + _______
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
3
Lesson 11
COMMON CORE MATHEMATICS CURRICULUM
M5
GEOMETRY
Definition:
A circle is a locus (set) of points in a plane equidistant from a fixed point.
Circle whose center is at (h,k)
Circle whose center is at the origin
(This will be referred to as the "center-radius form".
It may also be referred to as "standard form".)
Equation:
Equation:
Example: Circle with center (0,0), radius 4 Example: Circle with center (2,-5), radius 3
Graph:
Graph:
Now, if we "multiply out" the above example
we will get:
When multiplied out, we obtain the
"general form" of the equation of a circle. Notice
that in this form we can clearly see that the
equation of a circle has both x2 and y2 terms and
these terms have the same coefficient (usually 1).
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
4
Lesson 11
COMMON CORE MATHEMATICS CURRICULUM
M5
GEOMETRY
When the equation of a circle appears in "general form", it is often beneficial to convert
the equation to "center-radius" form to easily read the center coordinates and the radius for
graphing.
Examples:
1. Convert
form.
into center-radius
This conversion requires use of the technique of completing the square.
We will be creating two perfect square trinomials within the equation.
• Start by grouping the x related terms
together and the y related terms
together. Move any numerical constants (plain
numbers) to the other side.
• Get ready to insert the needed values for
creating the perfect square
trinomials. Remember to balance both sides of
the equation.
• Find each missing value by taking half of the
"middle term" and squaring. This value will
always be positive as a result of the squaring
process.
• Rewrite in factored form.
You can now read that the center of the circle is at (2, 3) and the radius is
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
.
5
Lesson 11
COMMON CORE MATHEMATICS CURRICULUM
M5
GEOMETRY
Determine the center and radius of the following circle equation:
x2+y2−8x−20y+107
=
0
MOVE
x2+y2−8x−20y
= −107
CONSTANT TO
OTHER SIDE
x2−8x+y2−20y
= −107
REARRANGE
x magic number=(−8/2) =(−4)2=16
y magic number=(−20/2) =(−10)2=100
x2−8x+16+y2−20y+100=−107+16+100−−−−−−−−−−−−−
(x2−8x+16)+( y2−20y+100)
=
9−
ADD MAGIC
NUMBERS
GROUP AND
ADD NUMBERS
BRING
(x−4)2+(y−10)2=9
CONSTANT TO
OTHER SIDE.
Center: (4,10) (change signs) and Radius: √9=3
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
6
COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
M5
GEOMETRY
Example 1
The following is the equation of a circle with radius 5 and center (1, 2). Do you see why?
𝑥 2 − 2𝑥 + 1 + 𝑦 2 − 4𝑦 + 4 = 25
Exercise 1
1.
Rewrite the following equations in the form (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2 .
a.
𝑥 2 + 4𝑥 + 4 + 𝑦 2 − 6𝑥 + 9 = 36
b.
𝑥 2 − 10𝑥 + 25 + 𝑦 2 + 14𝑦 + 49 = 4
Example 2
What is the center and radius of the following circle?
𝑥 2 + 4𝑥 + 𝑦 2 − 12𝑦 = 41
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
7
COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
M5
GEOMETRY
Exercises 2–4
2.
Identify the center and radius for each of the following circles.
a.
𝑥 2 − 20𝑥 + 𝑦 2 + 6𝑦 = 35
b.
𝑥 2 − 3𝑥 + 𝑦 2 − 5𝑦 =
19
2
3.
Could the circle with equation 𝑥 2 − 6𝑥 + 𝑦 2 − 7 = 0 have a radius of 4? Why or why not?
4.
Stella says the equation 𝑥 2 − 8𝑥 + 𝑦 2 + 2y = 5 has a center of (4, −1) and a radius of 5. Is she correct? Why or
why not?
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
8
COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
M5
GEOMETRY
Exercise 5
5.
Identify the graphs of the following equations as a circle, a point, or an empty set.
a.
𝑥 2 + 𝑦 2 + 4𝑥 = 0
b.
𝑥 2 + 𝑦 2 + 6𝑥 − 4𝑦 + 15 = 0
c.
2𝑥 2 + 2𝑦 2 − 5𝑥 + 𝑦 +
Lesson 11:
Date:
13
=0
4
Recognizing Equations of Circles
3/15/15
9
COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
M5
GEOMETRY
Summary
6.
The graph of the equation below is a circle. Identify the center and radius of the circle.
𝑥 2 + 10𝑥 + 𝑦 2 − 8𝑦 − 8 = 0
7.
Describe the graph of each equation. Explain how you know what the graph will look like.
a.
𝑥 2 + 2𝑥 + 𝑦 2 = −1
b.
𝑥 2 + 𝑦 2 = −3
c.
𝑥 2 + 𝑦 2 + 6𝑥 + 6𝑦 = 7
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
10
COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
M5
GEOMETRY
Problem Set
1.
Identify the centers and radii of the following circles.
a.
(𝑥 + 25) + 𝑦 2 = 1
b.
𝑥 2 + 2𝑥 + 𝑦 2 − 8𝑦 = 8
c.
𝑥 2 − 20𝑥 + 𝑦 2 − 10𝑦 + 25 = 0
d.
𝑥 2 + 𝑦 2 = 19
e.
𝑥 2 + 𝑥 + 𝑦2 + 𝑦 = −
Lesson 11:
Date:
1
4
Recognizing Equations of Circles
3/15/15
11
COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
M5
GEOMETRY
2.
Sketch graphs of the following equations.
a.
𝑥 2 + 𝑦 2 + 10𝑥 − 4𝑦 + 33 = 0
b.
𝑥 2 + 𝑦 2 + 14𝑥 − 16𝑦 + 104 = 0
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
12
COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
M5
GEOMETRY
c.
𝑥 2 + 𝑦 2 + 4𝑥 − 10𝑦 + 29 = 0
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
13
COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
M5
GEOMETRY
3.
Chante claims that two circles given by (𝑥 + 2)2 + (𝑦 − 4)2 = 49 and 𝑥 2 + 𝑦 2 − 6𝑥 + 16𝑦 + 37 = 0 are externally
tangent. She is right. Show that she is right on the graph below.
Lesson 11:
Date:
Recognizing Equations of Circles
3/15/15
14