Equations of Circles LESSON 10-4 Created by Lisa Palen and Kristina Green Henrico High School

Equations of Circles

LESSON 10-4

Created by Lisa Palen and Kristina GreenHenrico High School

Equations of Circles

Part I

Recall: Definitions

• Circle: The set of all points that are the same distance from the center

• Radius: a segment whose endpoints are the center and a point on the circle

• Radius: the LENGTH of a radius

RadiusCenter

Equation of a Circle2 2 2x y r Center (0, 0)

Radius = r

Center (h, k)Radius = r 2 2 2x h y k r

Finding the Center and the Radius when given the equation

2 2

2 2

2 2

2 2

22

2 2

1. 25

2. 100

3. 5 4 49

4. 7 3 3

5. 1 12

6. 3 81

x y

x y

x y

x y

x y

x y

Center (0, 0), r = 5Center (0, 0), r = 10

Center (5, -4), r = 7

Center (-7, 3), r =

Center (0, 1), r =

Center (3, 0), r = 9

3

12

Writing the Equation of a Circle

1. Center (0, 0) r = 2

2. Center (0, 1) r = 6

3. Center (-3, 5) r = 2.5

4. Center (-5, 10) r = 10

5. Center (8, 0) r = 1

6. Center (6, 9) r = 3.4

x2 + y2 = 4x2 + (y – 1)2 = 36(x + 3)2 + (y– 5)2= 6.25(x + 5)2 + (y–10)2= 100(x – 8)2 + y2= 1(x– 6)2 + (y– 9)2= 11.56

Writing the Equation of a circle2. A circle whose center is at (-3, 2) passes through (-7, 2).

a. What is the length of the radius of the circle?

b. Write the equation of the circle.

Answers: a. r = 4 b. (x + 3)2 + (y - 2)2 = 16

Graphing a CircleFind the center and the radius and graph the circle.

2 2 9x y

Answers: center (0, 0) radius = 3


2 21 2 25x y

Answers: center (1, -2) radius = 5


2 23 4x y

Answers: center (3, 0) radius = 2

Writing the Equation of a circle3. A circle has a diameter with endpoints

A (1, 2) and B (3, 6).

a. What is the center of the circle?

b. What is the radius of the circle?

c. What is the equation of the circle?Answers: a. (2, 4) b. sqrt (5) c. (x – 2)2 + (y – 4)2 = 5

The midpoint of segment AB!

The distance from the center to A or B!

diam

eter

Finding the midpoint

For the last problem it was necessary to find the midpoint, or the point halfway between two points. There is a formula for this.

Midpoint

Part II

Reminder: What is a Midpoint?• The midpoint of a segment AB is the point that divides

AB into two congruent segments.• Where is the midpoint of AB?

A

BOver Here

?

Over Here

?

Over Here

?

Here it is!

midpoint

Midpoint on a Number Line

• To find the midpoint of two points on a number line, just average the coordinates.

• Find the midpoint of GT.

a b

2

x

• Take the average of the coordinates:

G T

4 9

2

5

2 = 2.5

midpoint

a b

2

Finding a Midpoint inThe Coordinate Plane

x

y

We can find the midpoint between any two points in the coordinate plane by finding the midpoint of the x-coordinates and the midpoint of the y-coordinates.

midpoint?

Example Find the midpoint of the two points.


a b 2 3 10.5

2 2 2

x

y

First: Find the average (midpoint) of the x-coordinates.Remember: Take the average of the two coordinates.

– 4

8

average of x-coordinates

a b 4 8 42

2 2 2

2


a b 2 3 10.5

2 2 2

x

y

Next: Find the midpoint (average) of the y-coordinates.Remember: Take the average of the two coordinates.

– 2

3

a b 2 3 10.5

2 2 2

average of y-coordinates

average of x-coordinates

0.5

2


a b 2 3 10.5

2 2 2

x

y

midpoint of y-coordinates

midpoint of x-coordinates

0.5

2

Finally: The midpoint is the ordered pair:

(average of x-coordinates, average of y-coordinates)

= (2, 0.5)

(2, 0.5)

The Midpoint FormulaThe following formula combines what we did:

midpoint = (average of x-coordinates, average of y-coordinates)

where (x1, y1) and (x2, y2) are the ordered pairs corresponding to the two points.

So let’s go back to the example.

1 2 1 2x x y y,

2 2

Example

x

y

Find the midpoint of the two points.

Solution: We already know the coordinates of the two points.

(– 4, – 2)

(8, 3)midpoint?

Example cont.

1 2 1 2x x y y,

2 2

4 8 2 3,

2 2

4 1,

2 2

Solution cont.

Since the ordered pairs are

(x1, y1) = (-4, -2) and (x2, y2) = (8, 3)

Plug in x1 = -4, y1 = -2, x2 = 8 and y2 = 3 into

midpoint =

=

=

= (2, 0.5)

THINK ABOUT IT

Find the center, the length of the radius, and write the equation of the circle if the endpoints of a diameter are (-8,2) and (2,0).

Center: Use midpoint formula!

Length: use distance formula with center and an endpoint

8 2 2 0,

2 2

3,1 2 2(2 ( 3)) (0 1) 26

Equation: Put it all together

22 2( 3) ( 1) 26x y or 2 23 ( 1) 26x y

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Equations of Circles LESSON 10-4 Created by Lisa Palen and Kristina Green Henrico High School