Circles:Objectives/Assignment

Write the equation of a circle. Use the equation of a circle and its

graph to solve problems. Graphing a circle using its four

quick points.

CIRCLES

What do you know about circles?

Definitions

Circle: The set of all points that are the same distance (equidistant) from a fixed point.

Center: the fixed points Radius: a segment whose

endpoints are the center and a point on the circle

RadiusCenter

The equation of circle centered at (0,0) and with radius r

x 2 + y 2 = r 2

Solution:Let P(x, y) represent any point on the

circle

1

0.5

-0.5

-1

-1.5

-2 -1 1 2

yx

PP

P

P

Finding the Equation of a Circle

The center is (0, 0) The radius is 12

The equation is: x 2 + y 2 = 144

Write out the equation for a circle centered at (0, 0) with radius =1

Solution:Let P(x, y) represent any point on the

circle

122 yx

Equation of a Circle

2 2 2x h y k r

Center : ,h k

Radius : r

Ex. 1: Writing a Standard Equation of a Circle centered at (-4, 0) and radius 7.1

(x – h)2 + (y – k)2 = r2 Standard equation of a circle.

[(x – (-4)]2 + (y – 0)2 =7.12 Substitute

values.

(x + 4)2 + y2 = 50.41 Simplify.

Ex. 2: Writing a Standard Equation of a Circle The point (1, 2) is on a circle whose center is (5,-1).

Use the Distance Formula

Substitute values.

Simplify

Addition Property

Square root the result.

5

25

916

)3(4

)21()15(

22

22

22

r

r

r

r

r

yxr

The equation of the circle is:

(x –5)2 +[y –(-1)]2 = 52 (x –5)2 +(y + 1)2 = 25

Simplify

You try!!: Write the equation of the circle :

Center : Radius len( 3,6) 5gth:

Center : Radius len(5, 4) 7gth:

(x-5)2 + ( y+4)2 = 49

(x+3)2 + ( y- 6)2 = 25

THINK ABOUT ITFind 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 radius and an endpoint8 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

Finding the Equation of a Circle

Circle A

The center is (16, 10)

The radius is 10

The equation is (x – 16)2 + (y – 10)2 = 100

Finding the Equation of a Circle

Circle B

The center is (4, 20)

The radius is 10

The equation is (x – 4)2 + (y – 20)2 = 100

Finding the Equation of a Circle

Circle O

The center is (0, 0)

The radius is 12

The equation is x 2 + y 2 = 144

Graphing Circles If you know the equation of a

circle, you can graph the circle by identifying its center and

radius; By listing four quick points: the

upmost, lowest, leftmost and rightmost points.

Graphing Circles Using 4 quick points

(x – 3)2 + (y – 2)2 = 9

Center (3, 2)Radius of 3Leftmost point (0, 2)Rightmost point(6, 2)Highest point(3, 5)Lowest point(3, -1)

Graphing Circles by listing its four quick points

(x + 4)2 + (y – 1)2 = 25

Center (-4, 1)Radius of 5Left: (-9, 1)Right: (1, 1)Up: (-4, 6)Low: (-4, -4)

Ex. 4: Applying Graphs of Circles

A bank of lights is arranged over a stage. Eachlight illuminates a circular area on the stage. A coordinate plane is used to arrange the lights, using the corner of the stage as the origin. The equation (x – 13)2 + (y - 4)2 = 16 represents one of the disks of light.

A. Graph the disk of light.

B. Three actors are located as follows: Henry is at (11, 4), Jolene is at (8, 5), and Martin is at

(15, 5). Which actors are in the disk of light?

Ex. 4: Applying Graphs of Circles 1. Rewrite the equation to find the center

and radius. – (x – h)2 + (y – k)2= r2 – (x - 13)2 + (y - 4)2 = 16 – (x – 13)2 + (y – 4)2= 42 – The center is at (13, 4) and the radius is

4. The circle is shown on the next slide.

Ex. 4: Applying Graphs of Circles

Graph the disk of light

The graph shows that Henry and Martin are both in the disk of light.

Ex. 4: Applying Graphs of Circles

A bank of lights is arranged over a stage. Each light is

5

25

916

)3(4

)21()15(

22

22

r